The development is the latest sign of an apparent rift between Bercow and May. The speaker had already incensed the prime minister’s office by pointing to a 1604 legal precedent as justification for blocking May’s hopes for a third vote on her proposed withdrawal agreement, which had already been voted down twice.

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May returned to Brussels and was offered a short extension of the deadline for Britain to leave the European Union, which had been March 29. Parliament hopes to vote again on May’s deal next week, though it is unclear how this can proceed without Bercow’s approval.

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Speaking in front of No. 10 Downing Street on Wednesday evening, the prime minister pointed a finger at British lawmakers as culpable for the delay.

“This delay is a matter of great personal regret for me. You, the public, have had enough,” May said. “You want this stage of the Brexit process to be over and done with. I agree — I am on your side.”

She asked of the House of Commons: “Do they want to leave the E.U. with a deal? Do they want to leave without a deal? Or do they not want to leave at all? So far, Parliament has done everything possible to avoid making a choice.”

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The comments were viewed as a potentially risky new strategy by British analysts.

“Theresa May has pitched herself tonight against Parliament on the side of the people,” Laura Kuenssberg, the BBC’s political editor, wrote for the broadcaster’s website. “On her own side, some MPs have openly questioned the merit of her evening at the podium — toxic and delusional are some of the descriptions given.”

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Wes Streeting, a Labour lawmaker, said May’s message could whip up anger toward members of Parliament, some of whom already have received death threats.

He called May’s speech “incendiary and irresponsible. If any harm comes to any of us, she will have to accept her share of responsibility.”

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A Downing Street spokeswoman told reporters that the prime minister’s office “flatly” rejected claims that May’s statement put lawmakers at risk.

But lawmakers said the rhetoric hurts May’s cause.

“There’s absolutely no chance she is going to win over MPs in sufficient numbers after that statement,” Lisa Nandy, another Labour lawmaker, told the ITV broadcaster. “It was an attack on liberal democracy itself. . . . I will not support a government that takes such a reckless, dangerous approach.”

In Parliament on Thursday, a number of lawmakers expressed concern about the comments. Pete Wishart, a lawmaker with the Scottish National Party, said it was “the height of irresponsibility” for May to make the comments in the tense atmosphere.

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With May in Brussels, it was Andrea Leadsom, a member of May’s Conservative government and leader of the House of Commons, who sparred with Bercow and others in Parliament.

After Labour’s John Cryer said that the prime minister’s address was one of the most “contemptuous” he had ever seen and asked whether Leadsom agreed with it, Bercow stepped in and told an unnamed Conservative lawmaker to “grow up” and called on lawmakers to “raise the level.”

“Mr. Speaker, may I just say that your response does not raise the level. But I’ll leave it there,” Leadsom said in response.

“My response sets out the constitutional position that has applied to members of the House of Commons over generations,” Bercow said. “And I cannot for the life of me see or believe there is anything remotely controversial about what I have said.”

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The debate over the tone of the debate surrounding Brexit and the potential threats against the safety of lawmakers has been growing for years. In November 2016, the right-leaning tabloid newspaper Daily Mail was criticized after publishing a front-page story titled “Enemies of the People” that focused on three judges who had ruled that the government would require the consent of Parliament for their Brexit negotiations.

Less than a week ahead of Britain’s referendum on leaving the European Union, a member of Parliament for Labour, Jo Cox, was shot and stabbed to death by a man who held far-right views.

Karla Adams in London contributed to this report.