Theresa May has denied putting MPs in danger with her incendiary claim that they are responsible for the Brexit crisis and the delay to leaving the EU.

The prime minister’s spokesperson was forced to defend her TV statement, after warnings that the safety of MPs – some of whom have already suffered death threats – was now in greater jeopardy.

“I flatly reject that... there’s no evidence to suggest that,” she told journalists at a regular briefing.

Explaining the purpose of the speech, the spokesperson added: “What the prime minister was doing was setting out to the public that, while we are not going to be able to leave with a deal on time on 29 March, she is not prepared to support an extension beyond 30 June.

“It is three years since the referendum and the public want us to bring this to a conclusion and that is what she was setting out yesterday.”

The protests were led by Labour MP Wes Streeting, who said Ms May would be partly responsible if MPs came to harm following her “incendiary and irresponsible” speech.

He tweeted: “I’ve thought long and hard before saying this, but @theresa_may knows that MPs across the House are subjected to death threats – some very credible.

“Her speech was incendiary and irresponsible. If any harm comes to any of us, she will have to accept her share of responsibility.”

There were further protests in the Commons, including from fellow Labour MP Paula Sherriff who said she had previously “begged” Ms May to “dial down” her rhetoric.

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“Last week, I received a message saying my head should be chopped off, along with lots and lots of other messages,” she told parliament.

“I apprehended the prime minister last Thursday evening and I begged her, ‘Dial down the hate, prime minister, it is in your power to dial down the hate.’

“People are frightened, not just in this place but in the country as a whole and the prime minister must show some leadership. It is in her grasp.”

Diana Johnson, another Labour MP, said: “In recent weeks, like many MPs in this house, I have been accused of being a traitor. I’ve also had Facebook posts, along with two other MPs in Hull, saying we should be shot and hanged.”

John Bercow, the Commons speaker, put himself firmly on the side of MPs, saying: “None of you is a traitor, all of you are doing your best.”

The backlash came after Ms May delivered a “message to the public” on Wednesday night, telling viewers: “Of this, I am absolutely sure: you, the public, have had enough.

“You are tired of the infighting, you’re tired of the political games and the arcane procedural rows, tired of MPs talking about nothing else but Brexit when you have real concerns about our children’s schools, our National Health Service, knife crime.”