Brexit talks broke up in Brussels without a deal today, after a proposed solution for the Irish border met fierce resistance from the Democratic Unionist Party.
Discussions are set to resume later this week, with both Prime Minister Theresa May and European Commission president Jean-Claude Juncker declaring themselves "confident" that a solution can be found in time for a key summit of the European Council on December 14.
Mrs May began a lunchtime meeting with the Commission president with hopes high that Brussels would be able to declare that sufficient progress had been made on the so-called "divorce issues" in order to allow the leaders of the remaining 27 EU nations to give the green light for trade talks to begin next week.
But reports that Mrs May was on the verge of agreeing a deal on "regulatory alignment" between Northern Ireland the Republic led the DUP to warn it would not back any agreement which threatened the territorial integrity of the UK.
Mr Juncker said the meeting was "friendly and constructive".
He went on: "I have to say that she's a tough negotiator, and not an easy one, and she's defending the point of view of Britain with all the energy we know she has, and this is the same on the side of the European Union.
"Despite our best efforts and significant progress we and our teams have made over the past days on the three main withdrawal issues, it was not possible to reach a complete agreement today.
"We now have a common understanding on most relevant issues, with just two or three open for discussion.
"These will require further consultation, further negotiation and further discussions.
"We stand ready to resume the negotiations with the United Kingdom here in Brussels later this week.
"But I have to say that we were narrowing our positions to a huge extent today, thanks to the British Prime Minister, thanks to the willingness of the European Commission to have a fair deal with Britain.
"I'm still confident that we can reach sufficient progress before the European Council of December 15.
"This is not a failure, this is the start of the very last round. I'm very confident that we will reach an agreement in the course of this week."
Mrs May said: "We have had a constructive meeting today. Both sides have been working hard in good faith.
"We have been negotiating hard. And a lot of progress has been made. And on many of the issues there is a common understandng.
"And it is clear, crucially, that we want to move forward together. But on a couple of issues some differences do remain which require further negotiation and consultation.
"And those will continue, but we will reconvene before the end of the week and I am also confident that we will conclude this positively."
DUP leader Arlene Foster spoke out after Ireland's deputy premier Simon Coveney said the Dublin Government's concerns over the post-Brexit border with Northern Ireland were set to be addressed fully.
Regulatory alignment could mean both Ireland and Northern Ireland following the same rules governing trade, to ensure that goods can continue to move freely across a "soft" border with no checks.
But critics say that it would effectively move the customs border between the UK and the Republic into the Irish Sea.
How to maintain a soft Irish border has emerged as the key sticking point in Brexit negotiations, after London indicated it was ready to up its offer on the so-called "divorce bill" to as much as £50 billion.
Speaking ahead of Mrs May's meeting with Mr Juncker, Mr Coveney told RTE Radio One he believed that the post-Brexit border would be "invisible" with "no barriers" and "will look very much like it looks today".
But Mrs Foster made clear the DUP would oppose the deal if it meant Northern Ireland being subjected to different rules from the rest of the UK.
Speaking at Stormont, she said: "We have been very clear. Northern Ireland must leave the EU on the same terms as the rest of the United Kingdom.
"We will not accept any form of regulatory divergence which separates Northern Ireland economically or politically from the rest of the United Kingdom.
"The economic and constitutional integrity of the United Kingdom will not be compromised in any way."
The DUP, which props up Mrs May's minority Government in the House of Commons, has previously warned it could withdraw its support in Westminster if a deal is proposed which threatens the integrity of the United Kingdom.
Mrs Foster's intervention disrupted the choreography of events in Brussels, as the PM broke off from talks for urgent telephone discussions with the DUP leader.
Mrs May's planned meeting with European Council president Donald Tusk was delayed, as was an expected statement by Irish Taoiseach Leo Varadkar.
Meanwhile the leaders of devolved governments in Scotland, Wales and London added a further complication by announcing that if Northern Ireland was to be offered a special status after Brexit, other parts of the UK should be offered a similar opportunity.
Scottish First Minister Nicola Sturgeon, who opposes Brexit, said: "If one part of UK can retain regulatory alignment with EU and effectively stay in the single market - which is the right solution for Northern Ireland - there is surely no good practical reason why others can't."
And Welsh First Minister Carwyn Jones said: "We cannot allow different parts of the UK to be more favourably treated than others.
"If one part of the UK is granted continued participation in the single market and customs union, then we fully expect to be made the same offer."
London's Labour mayor Sadiq Khan said the deal reportedly being discussed in Brussels would have "huge ramifications for London", which voted overwhelmingly to remain in the EU.
"If Theresa May has conceded that it's possible for part of the UK to remain within the single market and customs union after Brexit ... and a similar deal here could protect tens of thousands of jobs," said Mr Khan.
In Westminster, Brexit Minister Steve Baker and Mrs May's chief of staff, Gavin Barwell, were expected to update Tory MPs on Brexit negotiations in Parliament, as Brexit supporters voiced concern about the progress of events in Brussels.
Former Ukip leader Nigel Farage called on Mrs May to "leave office now".
Regulatory alignment in Ireland would be "a bitter betrayal" of the 17.4 million people who voted for EU withdrawal, said Mr Farage, adding: "It is a concession too far, for it will lead to endless problems in Scotland and it damages the integrity of the United Kingdom."
And he said reports that Mrs May was ready to allow a role for the European Court of Justice in overseeing EU citizens' rights in post-Brexit Britain were "utterly unacceptable".
"Theresa May has got to go," said Mr Farage. "If we want to leave the EU, she's got to leave office now."
Liberal Democrat Brexit spokesman Tom Brake said: " This whole situation could have been solved by keeping the entire country in the single market and the customs union. Instead, Theresa May has cowered in front of her backbenchers and driven forward a reckless Brexit which risks destabilising the whole UK.
"As each day goes by, it becomes clearer that the best deal for everyone is to stay in Europe. The people of the UK must be given a vote on the deal and an opportunity to exit from Brexit."
The executive director of the Open Britain campaign against hard Brexit, James McGrory, said: "One of the few things that generates agreement on Brexit is that it must not be allowed in any way to compromise the integrity of our United Kingdom. It appears today, however, to be doing just that.
"There is a solution that would solve all of these problems for the Government, which is to keep the whole of the UK in the single market and the customs union. That would avoid a hard border in Ireland, ensure a level playing field for businesses across our islands, and protect trade with the EU, which buys almost half of everything we export."