The suspect, Esteban Santiago, had been receiving psychological treatment while living in Alaska, the Associated Press reports.
The news agency said it had spoken to the suspect's brother, Bryan Santiago, who told them the family had been alerted to his treatment by Esteban's girlfriend.
It is not known what he was being treated for.
Esteban Santiago was born in New Jersey, but grew up in Puerto Rico, his brother told AP.
Separately, Associated Press quoted a spokesperson for the Puerto Rico National Guard as saying Esteban had been deployed to Iraq for a year.
Governor: My heart goes out
APCopyright: AP
Governor Rick Scott (file photo)Image caption: Governor Rick Scott (file photo)
Governor Rick Scott said "whoever is responsible will be held accountable to the full extent of the law".
"My heart goes out to every family impacted," he said, adding that all law enforcement agencies were working together.
He also confirmed that he had spoken to President-elect Donald Trump and Vice-President-elect Mike Pence, but had not contacted President Barack Obama.
He cited his "personal relationship" with the incoming administration when asked why he had not contacted the sitting president.
Florida Senate considers airport gun restriction
Coincidentally, the Florida senate is currently considering bill 140 - which aims to expand areas where legal owners of firearms can carry their weapons.
Among the measures in the current draft is removing the restriction on legal carrying of firearms in airport passenger terminals, reports the Tallahassee Democrat.
Florida SenateCopyright: Florida Senate
Suspect has no connection to Canada - embassy
The Canadian embassy has confirmed that the named suspect, Esteban Santiago, has no apparent connection to the country and did not fly from Canada.
That supports Air Canada's previous statement that no passenger by that name flew on their service.
Bomb squad detonates 'suspicious package'
Passengers still at the airport are warned not to panic if they hear an explosion - authorities are planning the controlled detonation of a package.
There were reports that the suspect, who has been unofficially named as Esteban Santiago, had just arrived on an Air Canada flight and took his gun out of checked luggage.
But Air Canada issued a statement: "We can confirm that we have no record of such a passenger by that name, or checked guns, on any of our flights to Fort Lauderdale."
It also said none of its staff or passengers had been injured in the incident.
DC-based journalist Brody Logan, who earlier tweeted about his family members at the airport, posted photos he says were from a staff room where they took shelter.
This unverified video posted by Fox News shows how people ran in all directions on to the Fort Lauderdale airport tarmac, including what looks from the footage like a runway.
Army National Guard tells ABC News alleged perpetrator Esteban Santiago was honorably discharged about four months ago.
'Loading in bathroom'
Broward County Commissioner Chip LaMarca says on Facebook the gunman arrived on a flight from Canada with a checked gun in his bag.
Mr LaMarca said the suspect claimed his bag and went to the bathroom to load the gun before coming out and firing.
Senior officials have told ABC News the suspect flew from Anchorage to Minneapolis, and finally to Fort Lauderdale.
Hands up and taking cover
More pictures from the scene...
APCopyright: AP
Police question people evacuating from Fort Lauderdale-Hollywood International AirportImage caption: Police question people evacuating from Fort Lauderdale-Hollywood International Airport
APCopyright: AP
People with hands up at the airportImage caption: People with hands up at the airport
APCopyright: AP
People take cover at the airportImage caption: People take cover at the airport
Pandemonium on the tarmac
Witnesses reported that panicking ensued as the gunfire rang out.
People scattered, running outside on to the tarmac with little direction from the authorities or any sign of safety/emergency protocols.
APCopyright: AP
APCopyright: AP
APCopyright: AP
APCopyright: AP
APCopyright: AP
Gun in checked luggage?
Sheriff Israel is asked about an unverified report that the attacker was a passenger on an Air Canada flight and took his gun from checked luggage, before going into the bathroom to load it and open fire. Sheriff says he will not comment because the investigation is ongoing.
No second shooter
Sheriff Scott Israel says reports of a second gunman at the airport were unfounded.
"At this point, it looks like he acted alone," he said. "There is no active second shooter."
The sheriff says it is too early to say either way whether this is terrorism.
BreakingShooter unharmed
Sheriff Scott Israel says the gunman opened fire at 1pm local time, shooting at least 13 people, five fatally.
The suspect is in custody, unharmed, and is being interviewed by police.
Sheriff Israel says the scene is considered "fluid and active".
Flight chaos as planes turn around
Flight path tracker FlightRadar24 tweets the paths of several planes due at Fort Lauderdale.
Some have diverted, while others have done a complete turnabout, heading back to their origin.
Live Reporting
By Tom Geoghegan and Jude Sheerin
All times stated are UK
Thanks for joining us
We're halting our live coverage of the situation in Florida for the moment.
But our online team will bring you the latest on the Fort Lauderdale shooting in our main story on the topic.
To recap:
Read more:Five people shot dead by Florida gunman
Health officials say all injured are stable
Local health officials said that of the five people taken to hospital, only two required surgery, and all are in a stable condition.
Suspect discharged from Alaska guard in August
The Alaska Army National Guard has confirmed that the suspect received a general discharge in August 2016 for unsatisfactory performance.
He joined the Guard in November 2014, Lt Col Candis Olmstead told the Associated Press.
First photos of suspect published
American news outlets have started publishing photos of the man they say is suspect Esteban Santiago, 26.
But police have yet to release any official images, or confirm openly that the photographs in circulation match the suspect in custody.
Suspect 'received psychological treatment'
The suspect, Esteban Santiago, had been receiving psychological treatment while living in Alaska, the Associated Press reports.
The news agency said it had spoken to the suspect's brother, Bryan Santiago, who told them the family had been alerted to his treatment by Esteban's girlfriend.
It is not known what he was being treated for.
Esteban Santiago was born in New Jersey, but grew up in Puerto Rico, his brother told AP.
Separately, Associated Press quoted a spokesperson for the Puerto Rico National Guard as saying Esteban had been deployed to Iraq for a year.
Governor: My heart goes out
Governor Rick Scott said "whoever is responsible will be held accountable to the full extent of the law".
"My heart goes out to every family impacted," he said, adding that all law enforcement agencies were working together.
He also confirmed that he had spoken to President-elect Donald Trump and Vice-President-elect Mike Pence, but had not contacted President Barack Obama.
He cited his "personal relationship" with the incoming administration when asked why he had not contacted the sitting president.
Florida Senate considers airport gun restriction
Coincidentally, the Florida senate is currently considering bill 140 - which aims to expand areas where legal owners of firearms can carry their weapons.
Among the measures in the current draft is removing the restriction on legal carrying of firearms in airport passenger terminals, reports the Tallahassee Democrat.
Suspect has no connection to Canada - embassy
The Canadian embassy has confirmed that the named suspect, Esteban Santiago, has no apparent connection to the country and did not fly from Canada.
That supports Air Canada's previous statement that no passenger by that name flew on their service.
Bomb squad detonates 'suspicious package'
Passengers still at the airport are warned not to panic if they hear an explosion - authorities are planning the controlled detonation of a package.
Air Canada: No record of suspect or gun
There were reports that the suspect, who has been unofficially named as Esteban Santiago, had just arrived on an Air Canada flight and took his gun out of checked luggage.
But Air Canada issued a statement: "We can confirm that we have no record of such a passenger by that name, or checked guns, on any of our flights to Fort Lauderdale."
It also said none of its staff or passengers had been injured in the incident.
Passengers hid in back rooms
DC-based journalist Brody Logan, who earlier tweeted about his family members at the airport, posted photos he says were from a staff room where they took shelter.
His family are now safe, he said.
Video shows panic
This unverified video posted by Fox News shows how people ran in all directions on to the Fort Lauderdale airport tarmac, including what looks from the footage like a runway.
Suspect 'honorably discharged'
Army National Guard tells ABC News alleged perpetrator Esteban Santiago was honorably discharged about four months ago.
'Loading in bathroom'
Broward County Commissioner Chip LaMarca says on Facebook the gunman arrived on a flight from Canada with a checked gun in his bag.
Mr LaMarca said the suspect claimed his bag and went to the bathroom to load the gun before coming out and firing.
Senior officials have told ABC News the suspect flew from Anchorage to Minneapolis, and finally to Fort Lauderdale.
Hands up and taking cover
More pictures from the scene...
Pandemonium on the tarmac
Witnesses reported that panicking ensued as the gunfire rang out.
People scattered, running outside on to the tarmac with little direction from the authorities or any sign of safety/emergency protocols.
Gun in checked luggage?
Sheriff Israel is asked about an unverified report that the attacker was a passenger on an Air Canada flight and took his gun from checked luggage, before going into the bathroom to load it and open fire. Sheriff says he will not comment because the investigation is ongoing.
No second shooter
Sheriff Scott Israel says reports of a second gunman at the airport were unfounded.
"At this point, it looks like he acted alone," he said. "There is no active second shooter."
The sheriff says it is too early to say either way whether this is terrorism.
BreakingShooter unharmed
Sheriff Scott Israel says the gunman opened fire at 1pm local time, shooting at least 13 people, five fatally.
The suspect is in custody, unharmed, and is being interviewed by police.
Sheriff Israel says the scene is considered "fluid and active".
Flight chaos as planes turn around
Flight path tracker FlightRadar24 tweets the paths of several planes due at Fort Lauderdale.
Some have diverted, while others have done a complete turnabout, heading back to their origin.