In a televised interview this week, Mrs. May insisted that a mutually acceptable agreement could be reached, but also warned her many British critics that the alternative to anything she can negotiate could be nothing at all.

That could mean a so-called cliff-edge departure that would leave trucks marooned in ports, disrupt food and medical supplies and render factories idle.

On Wednesday evening, Mrs. May met with European leaders over dinner in Salzburg. On Thursday, leaders are likely to agree to a special meeting in November, which would set the latest in a series of slipping Brexit deadlines.

And, of course, getting to an agreement by then, even a vague one, will not be easy.

Of two accords that need to be reached, the more crucial and harder to fudge is a withdrawal treaty that includes backstop plans to avert new controls at the border between Northern Ireland, which is part of the United Kingdom, and Ireland, which will remain in the European Union.

If this can be agreed, it will trigger a standstill transition period until December 2020 during which little would change for Britons while the details of future trade ties would be thrashed out.

Those would be based on an agreement on future trading links, an issue that has long divided Mrs. May’s party.

In July, Mrs. May finally broke months of internal deadlock in a meeting in her official, 16th-century country residence, Chequers, and proposed keeping Britain aligned with European standards on goods and food, though not services. That prompted the resignation of two hard-line pro-Brexit cabinet ministers: Boris Johnson, the foreign secretary, who has since likened the Chequers plan to donning a suicide vest, and David Davis, who quit as Brexit secretary.