Jo Brand is being investigated by police over an allegation of incitement to violence after she joked about throwing battery acid over politicians.
The Metropolitan Police said they had received a complaint about the BBC Radio 4 comedy programme in which Brand made her comments.
It came as Theresa May asked the BBC to explain why it had approved the joke for broadcast, suggesting that it “normalised” violence against politicians.
Brand had been referring to the recent milkshake attacks on political candidates, including Nigel Farage, on the Heresy comedy show. She said: “Certain unpleasant characters are being thrown to the fore and they’re very, very easy to hate and I’m kind of thinking, why bother with a milkshake when you could get some battery acid?
“That’s just me. I’m not going to do it. It’s purely a fantasy. But I think milkshakes are pathetic, I honestly do, sorry.”
Mr Farage called her comments “incitement of violence” and said the police must act, although he claimed not to have made a complaint.
He told the Telegraph: “I’m going to ask Jo Brand and the BBC whether they would like to contribute to the costs of keeping me safe,” in a reference to the tens of thousands of pounds a year he spends on security.
“This is way above any norms of free speech - it is appalling and the fact that the BBC spends £177 million a year on light entertainment and comedy… this was a pre-recorded programme which they still chose to put out.”
He also described the comments as “completely and utterly disgusting. Can you imagine if I was to tell a story like that, about somebody on the other side of me, an Anna Soubry or someone like that?
“I reckon the police would knock on my door within 10 minutes. I think it’s appalling.”
Appearing at the Henley Literary Festival on Thursday, Brand apologised "for any offence caused" and said her comments had been edited and taken out of context.
Earlier, speaking outside her home in Dulwich, south-east London, she was asked if she would continue working with the BBC and responded: “I’m not employed by the BBC, so how can they sack me?”
When asked if she would continue working with the BBC, she responded: “I’m not employed by the BBC, so how can they sack me?” She added that freedom of speech is “extremely important”.
Appearing later at the Henley Literary Festival, Brand said she was “happy” with how she had phrased the joke.
A spokesman for the Prime Minister referred to Brendan Cox, whose wife, MP Jo Cox, was murdered in 2016.
“The Prime Minister has been repeatedly clear that politicians should be able to go about their work and campaign without harassment, intimidation or abuse.
“I note that Brendan Cox has said that violence and intimidation should not be normalised and we should consistently stand against it.
“The Prime Minister shares this view. It is for the BBC to explain why it considers this to have been appropriate content for broadcast,” the spokesman said.
Initially, the BBC offered a robust defence of Brand’s joke and it could still be heard on the BBC Sounds app on Thursday afternoon.
But after the police announced their investigation, the corporation appeared more contrite and the offending lines were edited out.
A BBC spokesman said: “Heresy is a long-running programme where, as the title implies and as our listeners know, panellists often say things which are deliberately provocative and go against societal norms but are not intended to be taken seriously.
“We carefully considered the programme before broadcast. It was never intended to encourage or condone violence, and it does not do so, but we have noted the strong reaction to it.
“Comedy will always push boundaries and will continue to do so, but on this occasion we have decided to edit the programme. We regret any offence we have caused.”
Heresy’s host, Victoria Coren Mitchell, continued to defend Brand’s words, saying it was “an edgy, naughty joke from a kind and funny woman.”
The Metropolitan Police said in a statement: "Police have received an allegation of incitement to violence that was reported to the MPS on 13 June.
"The allegation relates to comments made on a radio programme. The allegation is currently being assessed. There have been no arrests and inquiries are ongoing."