Environment minister George Eustice has quit the government over Theresa May's promise to allow MPs a vote on delaying Brexit, if her deal is rejected.
The MP said it would be "dangerous" to go to the EU "cap in hand at the 11th hour and beg for an extension".
He feared it could mean a long delay or that Brexit "may never happen at all" and said the UK must be prepared to walk away without a deal.
The PM said she was focused on leaving the EU with a deal on 29 March.
Mr Eustice is a longstanding Brexiteer, who stood as a UKIP MEP candidate before joining the Conservatives.
He told the BBC he would back the withdrawal deal the prime minister has negotiated with the EU, despite some reservations.
"I do think it's preferable to have an orderly Brexit and crucially, it's preferable to get Brexit done," he said.
"We have to get legally out of the European Union as quickly as possible within this window. If we don't and we end up with a long delay of two years, as some would like, then I really do fear we will be in a disastrous situation and Brexit may never happen at all."
Former Brexit Secretary Dominic Raab, who himself quit the government in November, suggested any delay to Brexit would reward the EU for its "intransigence" and reduce the chances of getting a deal.
"The issue with delay is at this point in time it weakens our leverage - why would the EU make concessions now?" he told BBC Radio 4's Today.
"I think from the EU's point of view it signals to them that actually their intransigence pays off and that's the wrong message for the UK to be sending to Brussels at this moment."
Mr Eustice, the MP for Camborne and Redruth in Cornwall, is the 14th member of Theresa May's government to resign over Brexit and said he was doing so with "tremendous sadness".
But in his resignation letter, he said he feared that the EU would end up "dictating the terms of any extension requested and the final humiliation of our country".
He added: "We cannot negotiate a successful Brexit unless we are prepared to walk through the door."
Fear of losing control
By the BBC's Deputy Political Editor John Pienaar
George Eustice resigned because he believes Mrs May's been manoeuvred into putting Brexit itself in doubt.
For him, the breaking point was allowing MPs to vote on whether to rule out a no-deal Brexit, he's one of many Brexiteers who are convinced the danger of a disruptive exit might add to the pressure on the EU to make concessions.
And he's especially upset about Mrs May promising a vote on whether to delay Brexit beyond 29 March, if only for a short time.
The prime minister was driven to volunteer those concessions by the fear of being defeated in the Commons this week, and having to concede them anyway.
Her de-facto deputy David Lidington, and Chief Whip Julian Smith, warned Mrs May plainly that she had no choice.
A core of ministers, senior, junior and their parliamentary aides, were willing to sacrifice their jobs if necessary to bring about that defeat.
She gave in, and hated doing so.
But the fear of George Eustice - shared by other Brexiteers is that once Brexit is delayed, the government loses control.
Mr Eustice's resignation comes after Theresa May's decision on Tuesday to allow MPs a vote on delaying the UK's departure from the EU, or ruling out a no-deal Brexit, if they again reject the withdrawal deal she has negotiated with the European Union.
The UK scheduled departure date from the EU is still 29 March - but that could be delayed if Theresa May fails to get her deal through Parliament in a vote she has promised will take place on or before 12 March.
In her reply, the prime minister said she was sorry he was resigning and praised his work as the longest serving minister at the Department for Food, Environment and Rural Affairs since the department was created, in 2001.
'Within our grasp'
She said: "I agree with you that Parliament must now come together and honour the referendum result by voting for a deal which will give businesses and citizens the certainty they need and deserve.
"Our absolute focus should be on getting a deal that can command support in Parliament and leaving on 29 March.
"It is within our grasp and I am grateful to have your continued support in that important mission."
Praising Mr Eustice on Twitter, his former boss Environment Secretary Michael Gove said he would be "very much missed".
So sorry to see George go. He has been a brilliant minister and will remain a dear friend. He leaves an outstanding legacy, with the Agriculture and Fisheries Bills setting domestic policy for the first time in nearly 50 years. He will be very much missed.— Michael Gove (@michaelgove) February 28, 2019
And former foreign secretary Boris Johnson praised the MP as "brave and right":
George Eustice is brave and right - and his superb letter to the PM shows that no deal may yet be the best option for the U.K.— Boris Johnson (@BorisJohnson) February 28, 2019
But the SNP's Westminster leader Ian Blackford said: "Another day, another resignation from the UK government. Any illusion to strong and stable ended before it began but this is beyond parody."
Who is George Eustice?
- Worked for his family's farming business in Cornwall before he entered politics
- Stood for UKIP in the 1999 European elections
- Director for the anti-euro "No" campaign
- Then Tory leader Michael Howard's head of press between 2003 and 2005
- Leading role in David Cameron's Tory leadership campaign and then his press secretary
- Elected as Conservative MP for Camborne and Redruth in 2010
- Junior environment minister in 2013 before joining Defra as a minister in 2013
- Campaigned to leave the EU ahead of the 2016 referendum, in which his constituency voted to leave by 58%
- Family still runs a fruit farm, restaurant and farm shop in Cornwall