Britain and the EU failed to strike a Brexit divorce deal after a dispute over the Irish border scuppered talks in Brussels on Monday, but both said they were confident of reaching an agreement later this week.
British Prime Minister Theresa May and European Commission President Jean-Claude Juncker appeared to have made a breakthrough on easing Dublin's concerns about future frontier arrangements.
Britain was prepared to keep the EU customs and single market rules for Northern Ireland in order to meet Dublin's insistence that Brexit should not bring back a "hard border" and threaten a peace process that ended decades of sectarian tensions on the island, sources said.
But any such deal was swiftly undone thanks to fierce opposition from Northern Irish unionists who prop up May's minority Conservative government, in a development that Ireland's premier called "surprising and disappointing".
The EU says Britain must make sufficient progress on key divorce issues -- Ireland, Britain's financial bill for leaving the bloc, and the rights of EU nationals in Britain -- to allow the opening of trade and transition talks at a summit on December 15.
"Despite our best efforts... it was not possible to reach a complete agreement today," Juncker said at a joint news conference with May, adding that she was a "tough negotiator."
"This is not a failure... I am very confident that we will reach an agreement in the course of this week," added the former Luxembourg premier.
May said differences remained on a "couple of issues".
"But we will reconvene before the end of the week, and I am also confident we will conclude this positively," May said.
EU President Donald Tusk -- who had earlier described Monday as the "absolute deadline" for a deal -- said that "it is now getting very tight but agreement at December (summit) is still possible."
Tusk, the former Polish premier who chairs EU summits, said he had been ready to present draft guidelines on Tuesday to start trade talks "but UK and (the) Commission asked for more time."
Northern Ireland's Democratic Unionist Party -- which has held the balance of power in the British parliament since May's disastrous showing in elections earlier this year -- said they rejected the deal despite a desperate telephone call from the British premier.
"We will not accept any form of regulatory divergence which separates Northern Ireland economically or politically from the rest of the United Kingdom," DUP leader Arlene Foster said in a statement.
Irish Prime Minister Leo Varadkar said he felt let down, having been asked by the EU chiefs whether he accepted a deal, only to hear later it had fallen through.
"I am surprised and disappointed that the British government now appears not to be in a position to conclude what was agreed earlier today," Varadkar said at a press conference in Dublin.
"I accept that the Prime Minister (Theresa May) has asked for more time, and I know that she faces many challenges, and I acknowledge that she is negotiating in good faith," he added.
In a sign of the tensions within the United Kingdom caused by Brexit, the leaders of Scotland and Wales together with the mayor of London all called for similar deals to the one being considered for Northern Ireland.
May, Brexit minister David Davis and the prime minister's Brexit adviser Olly Robbins attended the "working lunch" with Juncker, the EU Brexit negotiator Michel Barnier and Juncker's chief of staff Martin Selmayr.
The EU has demanded "sufficient progress" from Britain on the exit bill, citizens rights and Ireland in order to move on to talks on a post-Brexit transition period of up to two years, and on a future relationship including a trade deal.
Failure to do so this month could make the EU "rethink" whether an overall Brexit withdrawal deal is possible at all, Tusk has warned.
After months of stalemate, London and Brussels have effectively reached a deal on the divorce settlement, reported to be 45 billion to 55 billion euros ($51 to $63 billion), which was previously the most contentious issue.
A deal is also close on the rights of more than three million Europeans living in Britain, though there is still disagreement over whether they would be protected by the European Court of Justice -- a red line for Brexit supporters in Britain.