Last month, the High Court in London ruled that "as a matter of general principle, a suspect has a reasonable expectation of privacy in relation to a police investigation".
The judgment means the media will only be able to identify the individual in question if it can show that, on the facts of the case in question, its rights of freedom of expression and the public's right to know outweighs the privacy rights of the individual concerned.
The BBC unsuccessfully applied to the High Court for permission to appeal to the Court of Appeal. It subsequently considered lodging an appeal directly with the Court of Appeal but on Wednesday confirmed it would not do so.
In a statement it said: "The advice we have received is not promising. The legalities are complex, but essentially even though we are advised and believe that the judge erred in law in finding that broadcasters and journalists normally have no right to publish the name of a person who is the subject of a criminal investigation, it will be very difficult to persuade the Court of Appeal to isolate this issue of principle from the judge's broader findings in this case. The judgment has been written in a way that makes the two indivisible."
The BBC has, however, written to the attorney general in England and Wales, MP Geoffrey Cox QC, to request that the government look into updating legislation to address the restrictions it said the High Court's ruling has placed on press freedoms.
"This ruling will limit the long-standing ability of journalists to report on police investigations - many cases of which have resulted in further complainants coming forward," it said in its statement. "It will make it harder to scrutinise the conduct of the police and it will undermine the principle of the public’s right to know."
In its letter to Cox, the BBC said it accepted its reporting in the Sir Cliff case was "inappropriate" but reiterated its view that it was "entitled to report the fact of the investigation into his alleged conduct and to name him as the person concerned".
"The BBC believes that the probability is that the Court of Appeal, in upholding the judge's factual findings, would be very reluctant to address broader questions of principle," the BBC's letter said. "Indeed, the BBC is doubtful that an appeal on these specific facts is a sensible way of addressing the issues of principle. The court is very likely to say that it is for parliament, not the judiciary, to devise a statutory scheme setting out in detail the balance between competing public interests."
"For those reasons it may be more appropriate for the government to consider the merits of conducting a review of the state of the law on these issues, including an assessment of the need for primary legislation which will protect the right to report properly and fairly criminal investigations, and to name the person under investigation," it said.