SALZBURG, Austria — Britain and its European Union partners failed to secure a breakthrough in Brexit talks on Thursday, largely because of seemingly intractable divisions over the best way to avoid a hard border on the island of Ireland and how to deal with future trade.

With Britain's departure from the E.U. on March 29, 2019, looming, there are growing concerns that a deal on the post-Brexit relationship may not be cobbled together in time to ensure a smooth and orderly British exit.

Despite reports of a friendly spirit at a summit in Salzburg, Austria, the fundamental differences remained. E.U. Council President Donald Tusk said parts of British Prime Minister TheresaMay's Brexit plan — dubbed Chequers after a key Brexit meeting at the premier's country residence of the same name — simply "will not work."

But just minutes after he spoke, May insisted that her Brexit plan was the "only serious and credible" proposal on the table.

Tusk said he wants to see a major breakthrough by the time the leaders meet again in Brussels next month. He said a special Brexit summit could be held in mid-November if things progress as hoped — but only as a "punch line" if most of the deal had already been agreed.

If Britain is to leave with a deal in six months, May and the Europeans must find solutions in coming weeks so parliaments have enough time to ratify the agreement.

They spent two days in Salzburg trying to do just that, but with things at a standstill, each side tried to ramp up pressure on the other. Each is urging the other to compromise, while E.U. leaders issue constant warnings to Britain about the Brexit clock ticking.

"Time is running short," Irish Prime Minister Leo Varadkar told reporters. "We want to avoid a 'No Deal Brexit,' but we are preparing for that."

Tusk said that key parts of the British proposals to leave would threaten the union of the 27 remaining members, "not least because it risks undermining the single market" of seamless movement of goods, services, capital and persons.

May wants to keep the U.K. inside the bloc's single market for goods, but not services. The E.U. has insisted that the single market cannot be cherry-picked like that.

"Europe isn't an a la carte menu," French President Emmanuel Macron said.

The biggest single obstacle to a deal is the need to maintain an open border between the U.K.'s Northern Ireland and E.U. member Ireland. For Ireland, it's important not to undermine the hard-won peace after decades of sectarian tensions in neighboring Northern Ireland.

May said Britain and the E.U. agreed on the need for a legally binding backstop to guarantee there would be no hard Irish border. But Britain rejects the E.U.'s proposal, which would keep Northern Ireland inside the bloc's customs union while the rest of the U.K. leaves.

May said the backstop "cannot divide the United Kingdom into two customs territories." She said Britain "will be bringing forward our own proposals shortly" about how to break the impasse.

Dealing with the E.U. is only part of May's problem. Her Chequers plan also faces opposition from pro-Brexit members of her own Conservative Party, who say it would keep Britain tethered to the bloc, unable to strike new trade deals around the world.

Former Brexit Secretary David Davis, who quit the government in July, said Thursday that May's plan was "devoid of democracy" and "worse than no deal."

Macron, meanwhile, expressed contempt for pro-Brexit British politicians who told the public there would be "simple solutions" to leaving.

He said Brexit "demonstrated that those who explain that one can get by easily without Europe — that it will all go well, that it is easy and brings in a lot of money — are liars."

Any Brexit deal will include a withdrawal agreement and transition period to smooth Britain's exit from the bloc.

Currently that's expected to last until the end of 2020 — but without a deal Britain would crash out of the E.U. on Brexit day, a development that in theory could see flights parked and trade between the two sides grind to a halt.

It all suggests a fractious summit in Brussels next month.

"We are today at the moment of truth," Macron warned.