Since Britons voted in a 2016 referendum to leave the European Union, their divided government has struggled to define a negotiating position that reconciles conflicting objectives. While some want to keep close economic ties to the bloc to protect the economy, others would prefer a cleaner break and some hard-line pro-Brexit members of Mrs. May’s cabinet may be on the brink of resignation.

Talks broke down on Sunday, halting discussions over Brexit that many had hoped would reach a climax at the Brussels summit meeting. On Wednesday, Mrs. May was battling to save the negotiations and keep channels of communication open. She addressed the other leaders ahead of a dinner, insisting that a deal was possible before leaving them to their meal and their own discussion about the way ahead.

A summit meeting that had been penciled in for November to finalize a Brexit agreement is now unlikely to take place and most diplomats expect any deal will have to wait until December.

The big sticking point in the negotiations is the Irish border. On Wednesday, Mrs. May restated her commitment to ensuring that, whatever happens in talks on future trade ties between Britain and the European Union, there will be no need for new customs and other checks on goods at the frontier between Northern Ireland, which is part of the United Kingdom, and Ireland, which will remain in the European Union.

Given the difficulty of untying the knotty issue of how to manage a frictionless border between Northern Ireland and Ireland, most diplomats now expect that an agreement is unlikely before December. The European Union negotiator, Michel Barnier, called for calm discussions but said: “We need much more time.”