Few British institutions are as respected as the country’s health care provider, the National Health Service, and a pledge to divert money to it helped convince Britons to vote for withdrawal from the European Union.

But that claim was long ago debunked, and by promising a “Brexit dividend” for health provision on Monday, Prime Minister Theresa May heightened the acrimonious debate over Britain’s exit from the 28-country bloc while leaving key financial questions unanswered.

Speaking in London, Mrs. May said that the health service, which is under severe strain, would be given an extra $26 billion a year by 2023, an increase of around 3.4 percent.

“Some of the extra funding I am promising today will come from using the money we will no longer spend on our annual membership subscription to the European Union after we have left,” she said in a speech.