The UK and EU have agreed to carry on post-Brexit trade talks after a call between leaders earlier on Sunday.
In a joint statement, Boris Johnson and European Commission President Ursula von der Leyen said it was "responsible at this point to go the extra mile".
The pair discussed "major unresolved topics" during their call.
The two sides had said Sunday was the deadline for a decision on whether to continue with talks, with the UK set to leave EU rules at the end of the month.
The leaders agreed to tell negotiators to carry on talks in Brussels "to see whether an agreement can even at this late stage be reached".
They did not say how long these latest talks would continue, but the ultimate deadline is 31 December, and time must be allowed for the UK and European Parliaments to vote on any deal that emerges before then.
What are the sticking points in Brexit trade talks?
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What happens if there's no trade deal?
Mrs von der Leyen said Sunday's call with Mr Johnson had been "constructive and useful".
But Mr Johnson repeated his warning from earlier in the week that a no deal scenario was "most likely".
The UK and EU have been carrying out negotiations for a post-Brexit trade deal since March and are attempting to secure one before the so-called transition period end on 31 December - when the two sides would move to trading on World Trade Organization (WTO) rules.
Without a trade deal, tariffs - charges on goods being bought and sold between the two sides - could be introduced and, in turn, prices on certain products may go up.
Reading out the joint statement, Mrs von der Leyen said: "Despite the exhaustion after almost a year of negotiations, despite the fact that deadlines have been missed over and over, we think it is responsible at this point to go the extra mile."
Mr Johnson later said "where there is life, there is hope", and that the UK "certainly won't be walking away from the talks".
But he added: "I've got to repeat the most likely thing now is of course that we have to get ready for WTO terms.
"As far as I can see, there are some serious and very difficult issues that currently separate the UK from EU and the best thing to do now for everybody… [is to] get ready to trade on WTO terms."
Labour's Rachel Reeves welcomed the continuation of the talks and said the worst outcome would be to "crash out with no deal whatsoever on 1 January".
She added: "I hope that they [the talks] will swiftly conclude, but I also hope on behalf of all British businesses and workers, and our security as well, that the government deliver the promise they made to the British people and come back with a deal."
The basics
Brexit happened but rules didn't change at once: The UK left the European Union on 31 January 2020, but leaders needed time to negotiate a deal for life afterwards - they got 11 months.
Talks are happening: The UK and the EU have until 31 December 2020 to agree a trade deal as well as other things, such as fishing rights.
If there is no deal: Border checks and taxes will be introduced for goods travelling between the UK and the EU. But deal or no deal, we will still see changes.
What happens next with Brexit?
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Talks will now continue in Brussels, with a focus expected on how close the UK should stick to EU economic rules in the future.
The EU is determined to prevent the UK from gaining what it sees as an unfair advantage of having tariff-free access to its markets - not paying taxes on goods being bought and sold - while setting its own standards on products, employment rights and business subsidies.
The EU is reported to have dropped the idea of a formal mechanism to ensure both sides keep up with each other's standards and is now prepared to accept UK divergence - provided there are safeguards to prevent unfair competition.
Fishing rights is another major area of disagreement, with the EU warning that without access to UK waters for EU fleets, UK fishermen will no longer get special access to EU markets to sell their goods.
But the UK argues that what goes on in its own waters, and its wider business rules, should be under its control as a sovereign country.
Business lobby group the CBI said the continuation of talks "gives us hope", and that a deal was "both essential and possible" for the UK economy.
The National Farmers' Union has warned there will be "significant disruption" to the sector if the UK fails to reach a trade deal with the EU.
And the British Retail Consortium warned the public would face "over £3bn in food tariffs [meaning] retailers would have no choice but to pass on some of these additional costs to their customers".
Taoiseach (Irish prime minister) Micheál Martin said he believed a no-deal scenario "would be very bad news for all of us" and "an appalling failure of statecraft" on both sides.
Speaking to the BBC's Andrew Marr, he called for the teams "with any bit of energy we have left [to] focus on negotiating a deal".
Britons told not to stockpile food ahead of January
Where do UK political parties stand on Brexit now?
A number of Conservative MPs welcomed the continuation of talks, with former minister Damian Green, who backed Remain in the 2016 EU referendum, saying it was "good news" and that "no deal would be terrible".
But leading Tory Brexiteer Sir John Redwood tweeted: "A long complex legal agreement that locks the UK back into many features of the EU that hinder us is not the Christmas present the UK needs."
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