Magistrate Judge Enjolique Lett has ordered the US president to produce the relevant records by November 23, as she raised concerns over the collation of the paperwork by his legal team.
In the absence of a register identifying any documents which had been withheld due to claims of privilege, she also directed Trump to compile a list by November 5.
The court order signed by the judge said: "Failure to comply with this order may result in sanctions, including but not limited to, dismissal of this action for failure to prosecute this case."
Trump is seeking substantial damages in a Florida court over the BBC's editing of a Panorama documentary from 2024 about the US Capitol riots.
The BBC had previously accused the president of flouting the rules in failing to disclose documents as part of the case.
Given the legal timetable, lawyers for the BBC have argued it would "experience extreme prejudice" if Trump "continues to disregard his discovery obligations", and so had asked the court to intervene.
They pointed out the ongoing delay related to documents "that are not in dispute" and which Trump had previously agreed to disclose.
In her ruling, Judge Lett said: "After inquiry, plaintiff's counsel assured the court that all documents responsive to the RFPs (request for production) about which there remains no dispute were produced.
"However, plaintiff's counsel admitted that no privilege log identifying any documents that were withheld based on any asserted privilege has been produced.
"Given the nature of the requests, the paucity of the production, and the steps counsel undertook to ensure that all responsive documents were collected, the court has concerns about whether an adequate search for all responsive documentation occurred."
The move comes after the same judge recently ruled Trump must give an interview under oath by December 11, following complaints that he had delayed his pre-trial deposition, with the clock ticking on the time available to disclose evidence.
At a court hearing last week, a lawyer for the BBC, Charles Tobin, pointed out a date had been agreed upon of October 16, but Trump cancelled it due to a "matter of national importance".
Tobin said the president had actually been campaigning in Georgia, ahead of November's mid-term elections.
Trump is taking legal action over a Panorama episode, broadcast shortly before the 2024 presidential election he won.
The documentary has faced criticism for giving the impression Trump had encouraged his supporters to storm the Capitol building on January 6, 2021 after he lost the 2020 US election to Joe Biden.
The allegations led the then-BBC director-general Tim Davie to announce his resignation.
