Boris Johnson's dinner with EU chief Ursula von der Leyen - aimed at breaking the Brexit trade deadlock - has ended without agreement.
A No 10 spokesperson said "very large gaps remain" but talks will continue, with a "firm decision" by Sunday on whether a deal can be reached.
Mrs von der Leyen said the two sides were still "far apart".
Talks between the UK's chief negotiator Lord Frost and the EU's Michel Barnier will resume in Brussels later.
The two negotiators also attended the three-hour dinner meeting between the two leaders.
The BBC's Laura Kuenssberg said the evening had "plainly gone badly" and the chances of the UK leaving the post-Brexit transition period at the end of the year without a firm arrangement was a "big step closer".
Time is running out to reach a deal before 31 December, when the UK stops following EU trading rules.
Major disagreements remain on fishing rights, business competition rules and how a deal will be policed.
The dinner was seen as a last-ditch opportunity to work through the main sticking points and for the two sides to try and find some common ground.
If at first you don't succeed you can try and try. But eventually, sometimes failure is what follows.
That now seems the likely outcome of months of talks designed to create a smooth path for the country towards a different future - a deal that, in theory, would ease the junction from membership of a huge trading bloc to a world outside.
There is a chance still that a couple of frantic days could result in a change.
The prime minister could decide that after all, the potential disruption of no deal is just too great to risk.
The EU president might be able to persuade continental leaders to budge, as they gather in Brussels today.
But the chance of reassessing and refreshing the efforts seem now remote.
In a statement, the UK side said there had been "a frank discussion about the significant obstacles which remain in the negotiations".
"Very large gaps remain between the two sides and it is still unclear whether these can be bridged," a No 10 spokesperson said.
They said the two sides had agreed to further discussions over the next few days, and the PM did "not want to leave any route to a possible deal untested".
"The PM and VDL [von der Leyen] agreed that by Sunday a firm decision should be taken about the future of the talks," the spokesperson added.
Mrs von der Leyen said the discussions had been "lively and interesting", and the two sides fully "understand each other's positions" but they "remain far apart".
"We will come to a decision by the end of the weekend," she said.
Dinner between Boris Johnson and Ursula von der Leyen ended as predicted in Brussels - with neither a breakdown, nor a breakthrough in the trade talks impasse.
EU diplomats say the bloc is ready to go the extra mile during the next days of negotiations but contrary to the UK government view, the EU thinks the decision - deal or no deal - lies primarily in Downing Street.
Brexit isn't on the official discussion agenda at an EU summit starting in Brussels later today, though leaders will be briefed on the negotiations.
Attitudes seem to be hardening.
"No deal is better than a bad deal" is a sentiment you hear both sides of the Channel now.
The European Commission says it's about to publish what it calls 'very narrow' contingency plans to keep planes in the sky and goods trucks on the road, in case of no deal.
After the talks ended, MPs reacted to the news that there had not been any agreement.
"One year after Boris Johnson promised us an oven-ready deal he has completely failed," said Labour's deputy leader Angela Rayner. "The failure to deliver the deal he promised is his and his alone."
She was referring to the prime minister's promise of having an "oven-ready deal" at the last election. Earlier this week Mr Johnson said the oven-ready deal he was referring to was the Withdrawal Agreement, or divorce deal, rather than a trade deal.
SNP's Westminster leader Ian Blackford tweeted: "A no deal would be a massive failure of diplomacy and leadership which Boris Johnson has to take ownership of."
Meanwhile, Tory Brexiteer MP John Baron said the PM deserved praise for "standing firm" rather than compromising in a rush to agree a deal. "We must remember a trade deal is for keeps, not just for Christmas," he said.
"We all want a deal, but it has to be a good deal because as we've said many times before, no deal is better than a bad deal."
Speaking before he left for Brussels, Boris Johnson said the EU was insisting on terms "no prime minister could accept" in relation to access to UK fishing waters and retaliatory measures if the UK diverged from EU standards.
Brexit - 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.
Any deal also has to be ratified by the European Parliament and win the backing of MPs at Westminster.
The House of Commons could sit as late as Christmas Eve should it be required to look at a Brexit deal, Speaker Sir Lindsay Hoyle said.
Under current plans, the Commons will stop sitting on 21 December, but he told Sky News the recess could be delayed.
BBC
Russia's unmanned Luna-25 spacecraft has crashed into the Moon after spinning out of control, officials say.
Donald Trump has confirmed that he will not take part in Republican presidential debates with his rivals in the race for the White House.