The DUP has accused the Prime Minister of demeaning the strength of the UK as EU leaders last night looked set to offer a lengthy Brexit delay.
France and Germany were reportedly split on how long a delay Theresa May should be granted, with French President Emmanuel Macron initially opposing anything beyond June 30.
Most EU leaders at the emergency Brussels summit appeared ready to back European Council president Donald Tusk's proposal of a long extension to Article 50 talks.
But agreement was being held up by President Macron's stance.
Mrs May has asked for a delay to June 30, with a break clause allowing the UK to leave as soon as Parliament has ratified her withdrawal agreement.
She told MPs last month that "as Prime Minister" she would not delay Brexit beyond the end of June.
But arriving in Brussels yesterday she played down the significance of the comment, insisting that the important issue was securing the flexibility to leave early and avoid having to take part in European elections next month.
Tory sources said the Prime Minister stood by commitments made to the backbench 1922 Committee that she would hand over to a successor for the second phase of negotiations after ratification. But Mr Tusk's proposals could postpone the opening of the next phase, dealing with the future UK/EU trade and security relationship, to the spring of 2020. That could mean Mrs May remaining in power for another year.
She set out her case for a short extension in a one-hour presentation behind closed doors, during which she answered questions from the leaders of the 27 member states. She then left the room to allow them to discuss the UK's future in her absence over a dinner of scallop salad, loin of cod with brown shrimps and iced macadamia nut parfait.
Arlene Foster rounded on Mrs May for her handling of the Brexit negotiations.
The DUP leader said it was "outrageous" that almost three years after people voted to leave the EU, the UK seemed on course to hold another European election next month.
"The uncertainty we face is not the fault of the voters," she said.
"The Prime Minister should recognise that the decision to leave the EU is not the problem but rather the ham-fisted manner in which the negotiations have taken place."
Mrs Foster is set to meet the EU's chief Brexit negotiator Michel Barnier today.
She will be joined by former Secretary of State Owen Paterson, ex-Tory leader Iain Duncan Smith and the party's MEP Diane Dodds.
She accused the EU of having "consistently ignored" the views of unionists who don't want a new border erected between Northern Ireland and Britain.
"Despite the Prime Minister being warned about the opposition to her withdrawal agreement, she has limped along and tried to force people into a cul-de-sac where they have no option but to support her deal," Mrs Foster said.
"That is a weak approach and demeans the strength of this great nation.
"It is also foolish as it traps the UK and burdens future generations with a bad deal. Our great democratic principles have been damaged."
The DUP leader said the meeting with Mr Barnier would be an opportunity to set out her party's opposition to the withdrawal agreement and for him to then make changes.
Sinn Fein MEP Martina Anderson, who met Mr Barnier yesterday, told him there could be no hard border. "The British Government has shown its disregard for the Good Friday Agreement and the impact Brexit will have on Ireland," she said.
"Now as we enter into a crucial phase of the Brexit process, it is time for the EU and the Irish Government to stand up and ensure the Good Friday Agreement is protected and partition is not reinforced by Brexit."
Speaking as he arrived at the EU summit, Taoiseach Leo Varadkar said he was open to either a short or long extension to Article 50.
On Mrs May's statement that she was not prepared to stay in the EU past June 30, he said: "Ultimately that decision lies with the House of Commons because the UK can leave before June 30, it can leave well before June 30 - it can leave in a matter of weeks.
"All that's required is for the House of Commons to ratify the agreement.
"I hope and trust the cross-party talks will allow that outcome to continue and allow us to do what we're dying to do, really, which is to get busy negotiating a new future relationship between the UK and the EU, one that is a very strong economic and very strong security partnership."
Mr Varadkar expressed hope that the UK would join the EU in a customs union. "One thing I would like to see considered and which is under consideration is the possibility of a customs union being formed between the UK and EU," he added.