Theresa May has risked enraging Brexiteers by indicating she is ready to consider extending Britain's transition out of the EU for a further year to the end of 2021.
The Prime Minister told a crunch European Council summit in Brussels that she was "ready to consider" the extension floated by EU chief negotiator Michel Barnier to give time to resolve the intractable problem of the Irish border.
If agreed, the change would mean the UK remaining within the single market and customs union and subject to EU rules and regulations for almost three years after the official date of Brexit in March 2019 and more than five years after the referendum vote to Leave.
Former Ukip leader Nigel Farage said that it would delay full withdrawal almost until the general election scheduled for May 2022 and "may mean we never leave at all".
After her Chequers plan for Brexit was humiliatingly rejected at the last EU summit in Salzburg last month and efforts to seal a last-minute deal foundered last weekend over the EU's demand for a "backstop" to avoid a hard border in Ireland, Mrs May was fighting to keep the door open for an agreement to deliver an orderly withdrawal.
Addressing leaders of the 27 remaining EU member states in Brussels, she stressed that significant progress had been made in many areas of the negotiations and urged them to find a "creative" way out of the current dilemma.
"We have shown we can do difficult deals together constructively," she said.
"I remain confident of a good outcome."
The Prime Minister added: "The last stage will need courage, trust and leadership on both sides."
Following her 20-minute speech, Mrs May left to eat at the UK ambassador's residence in the Belgian capital, leaving the EU leaders to discuss Brexit in her absence over dinner.
The president of the European Parliament, Antonio Tajani, said: "Both sides mentioned the idea of an extension of the transition period as one possibility which is on the table and would have to be looked into."
A senior EU official later said Mrs May had indicated she was "ready to consider" a longer transition period.
Mrs May initially suggested an "implementation period" of around two years after Brexit to give the UK's authorities and companies time to prepare for the new arrangements.
But she later accepted a 21-month transition offered by the EU, ending on the last day of December 2020.
It emerged yesterday that Mr Barnier was ready to discuss a further year's extension to allow time to find a solution to keep the Irish border open.
UK officials stressed that the Prime Minister was not proposing any extension to the period already agreed.
Any further extension is likely to be fiercely opposed by Eurosceptics, who warn the UK would become a "vassal state" of Brussels, bound by its rules but unable to influence them.
And it is certain to involve a demand from Brussels for a further year's contributions towards EU budgets, which could cost the UK as much as £9 billion.
This week's European summit has long been billed as "the moment of truth" when agreement was needed to allow time for ratification before Brexit day in March.
But Mr Barnier made clear no breakthrough was now expected, saying that "much more time" was needed to bridge differences between the two sides, and promising to "continue the work in the next weeks calmly and patiently".
Mrs May did not come forward with the new "concrete proposals" on the border issue which European Council president Donald Tusk has said are needed to break the deadlock.
A number of EU leaders voiced their willingness to work for an orderly UK withdrawal.
But several also noted that their countries were beginning preparations for a possible no-deal Brexit.
In a speech to the German parliament before travelling to Brussels, Chancellor Angela Merkel said that the possibility of a Brexit deal was "still there", but added that Berlin was making plans for a no-deal withdrawal.
In Paris, Emmanuel Macron's government published details of legislation to authorise preparations for a no-deal Brexit, which could see the restoration of customs checks and health inspections for animals at French ports, and even a requirement for Britons to seek visas for stays of three months or more.
Mr Macron, who held separate talks with Mrs May ahead of the three-hour dinner, said it was time to "accelerate" talks.
But Lithuanian president Dalia Grybauskaite said there would be "no breakthrough" this week because Mrs May lacked a strong mandate from her party and parliament.
Britain needed "to decide finally what they want and to rally behind the Prime Minister all together, not split", she said, adding: "Today we do not know what they want.
"They do not know themselves what they want.
"It is a problem."