The 61-year-old comedian was appearing on Victoria Coren Mitchell's Heresy on Tuesday night and joked about throwing battery acid at politicians.
In reply to a question about the state of UK politics, she said: "Well, yes I would say that, but that's because 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."
In a tweet on Wednesday Mr Farage, who is the leader of the Brexit Party, accused Brand of inciting violence, although he did not say who against.
He wrote: "This is incitement of violence and the police need to act."
Last month Mr Farage was covered in milkshake during a campaign walkabout in Newcastle city centre.
He was heard to comment "complete failure" and "I could have spotted that a mile off" as he was ushered away by security following the incident.
The trend of throwing milkshakes at right-wing politicians began when viral footage showed Tommy Robinson having one thrown over him in Warrington.
Since then, several other members of the public have attempted to repeat the unusual protest in an apparent tribute to the original culprit.
At the end of show, Coren Mitchell said she hoped Brand's remarks had not caused offence, but added that the radio series had been set up to "test the boundaries of what it's OK to say and not say".
The quiz host and television personality, 46, later responded to Mr Farage on Twitter, accusing him of double standards.
She wrote: "Nigel! I'm genuinely disappointed; we don't agree on everything, but I would totally have had you down as a free speech man. Especially when it comes to jokes."
A BBC spokeswoman said: "Heresy is a long-running comedy 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."