British Prime Minister Theresa May faced down a call to resign on Thursday as she implored MPs to resolve the Brexit impasse at last after the EU granted a new six-month delay.

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May's 27 European Union counterparts pulled another all-nighter in Brussels before clinching a compromise timetable for Britain's departure, extending the deadline until October 31.

Britain will be able to leave before then if its parliament finally manages to ratify the hotly-contested divorce deal May reached with the bloc and that has been behind all the political turmoil in London, with MPs unable to back it -- or any alternative they themselves have come up with.

"The whole country is intensely frustrated that this process to leave the European Union has not still been completed," May told MPs in parliament.

She said lawmakers should reflect during their forthcoming Easter break on the decisions that will have to be made swiftly upon their return. "Let us then resolve to find a way through this impasse," said May.

"The choices we face are stark and the timetable is clear." The second departure extension -- Britain was otherwise due to crash out on Friday -- infuriated hardcore Brexiteers in her centre-right Conservative Party like William Cash, who want May's head.

They fear that the delay might be prolonged yet again -- and the extra time used to engineer a softer form of Brexit, or even see it annulled outright. Cash said May had presided over an "abject surrender" in Brussels, adding: "Will she resign?". I think you know the answer to that," she replied.

May has promised to step down once she delivers the first stage of Brexit -- meaning she could remain in power until near the end of the year. The delay avoids a possible economic calamity on both sides of the Channel but does little to resolve the political morass that has seen May's control over her MPs and cabinet gradually slip.

The pound traded steady and stocks held firm on news of the extension.

The delay allowed traders to breathe a sigh of relief but observers noted the reprieve was only brief with the agreement merely kicking the can down the road.

EU Council president Donald Tusk admitted in Brussels after the marathon meetings wound down that "everything is possible". "Our intention is to finalise the whole process in October... but I am too old to exclude another scenario," he told reporters.

May did get an unexpected boost from US President Donald Trump. "Too bad that the European Union is being so tough on the United Kingdom and Brexit," he tweeted.

France's Europe minister on Thursday defended President Emmanuel Macron's opposition to granting a Brexit extension of more than six months.

A long extension would have been seen "as an attempt by the EU to hamper Britain's departure by delaying the decision for as long as possible in hoping that they will change their mind," Amelie de Montchalin told parliament.

Unable to convince enough of her Conservatives and their Northern Irish allies to back the deal, May last week decided to try finding a compromise with the main opposition Labour Party.

A cross-party agreement could help her get a deal through by May 22, just in time for Britain to leave the EU without having to take part in European Parliamentary elections at the end of next month.

"Reaching an agreement will not be easy, because to be successful it will require both sides to make compromises," said May. Labour is primarily hoping to place Britain in a European customs union of some sort.

May had previously ruled this out because it prevents Britain from striking lucrative independent trade agreements with giants such as China and the United States.

Labour leader Jeremy Corbyn said the new delay represented a "diplomatic failure" and typified May's "mishandling of the entire Brexit process".

The veteran socialist said while the cross-party talks on finding a compromise were constructive and serious, "the red lines must move and we must see a real compromise".

In the working-class port town of Tilbury east of London, which strongly backed leaving the EU in the 2016 referendum, Brexit supporters voiced their frustration at the new delay. "I'm very disgruntled with it all," Suzy Hornsby, 59, told AFP. "We want out... I can't wait to leave."