The 77-year-old singer sued over BBC coverage of a South Yorkshire Police raid on his home in Sunningdale, Berkshire in August 2014, following a child sex assault allegation.
Earlier this month Mr Justice Mann ruled in Sir Cliff’s favour following a High Court trial in London.
The judge concluded coverage was a “very serious” privacy invasion and awarded Sir Cliff £210,000 damages.
He said the award would be made up of £190,000 to cover the “general effect” coverage had on Sir Cliff’s life, plus £20,000 because the BBC had aggravated harm by nominating their coverage for an award.
BBC bosses had to get permission to appeal by showing they have a realistic chance of overturning the ruling.
Either Mr Justice Mann or a Court of Appeal judge could give that permission.
Lawyers representing the BBC yesterday began that process at a follow-up hearing in London, asking Mr Justice Mann to give the BBC the go-ahead to mount an appeal. But the judge refused.
He said an appeal did not have a real prospect of success and added that there was no other compelling reason why Court of Appeal judges should consider the case.
Barrister Gavin Millar QC, who leads the BBC legal team, had said issues raised during the case meant there was a “compelling reason” for an appeal to be heard.
Mr Millar said the ruling had implications for journalists, and an appeal had a “real prospect” of success.
He also said the damages award was “wrong in law” and would have a “chilling effect”.
He added: “The risk is a severe chilling effect on the freedom of the press in relation to reporting police investigations.
“We don’t go as far as to say you passed a law.”
Lawyers representing Sir Cliff said the BBC should not get permission to appeal.
Barrister Justin Rushbrooke QC, who leads Sir Cliff’s legal team, said that Mr Justice Mann had applied the law to the facts.
“What your lordship has done is faithfully and painfully apply the law to the facts of this case,” he told the judge yesterday.
“It is about time the BBC took a realistic view of this matter.”
Sir Cliff denied the allegation. He was never arrested, and in June 2016 prosecutors announced he would face no charges.