More reporting from America suggesting it's unlikely in their courts:
https://archive.ph/CMxMR
The two biggest issues seem to be jurisdiction and convincing a court that his reputation was materially damaged by the report.
Quote:
First, it’s not clear that US courts even have jurisdiction, given that the documentary never aired in the US and was geo-blocked in the US on the BBC’s streaming service.
The president would also need to prove that the BBC had acted with “actual malice” toward him when it edited the documentary — a high bar required for public figures that was established by the US Supreme Court in 1964 in order to protect free speech.
“They’ve got to meet the actual malice standard with New York Times versus Sullivan, which is a very tough standard,” said Gregory Germain, a professor at Syracuse University College of Law. Trump is “the ultimate public figure.”
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