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Miss Brown was a person very similar to Peri Brown, but from another reality. She did not recognize the name Peri, insisting her name was "Miss Brown", and she similarly had no memories of travelling with the Fifth and Sixth Doctors. She also had a British accent, while Peri had an American one.
She encountered an alternate version of the Sixth Doctor on Aridius, informing him on the decisions made by the Fourth Doctor to alter the existence of the Daleks, creating a whole new timeline and another version of the Time War as a result. (AUDIO: Dust Devil)
Behind the scenes
The Stranger
Miss Brown seems to be a direct reference to the controversial Stranger series released by BBV Productions. The series featured Colin Baker and Nicola Bryant as the legally distinct characters the Stranger and Miss Brown. In the first three films, their names were never clarified via dialogue, and "the Stranger" was simply an on-screen credit. At one point, when Miss Brown is asked if her "friend" is a Doctor, she laughs as if there is some hidden truth to the statement.
The intention at the time was that the series was set during the Sixth Doctor's era, with any legal reference to the TV series being implied to happen off-screen. For instance, the characters travel in time and space to various planets, but it is never show how they do it, implying the TARDIS could be nearby. The third tape, In Memory Alone, featured both characters having amnesia to give credence to why they never mention any specific details in their lives. The only majour contradiction between Miss Brown and Peri Brown is that Miss Brown carries Nicola Bryant's natural British accent, contradicting Peri being an American.
Bryant would soon quit the films, and afterwards it would be retconned that the Stranger character was not the Doctor, but an interdimensional terrorist named Soloman. Soloman was a rebel leader lashing out against his own race, the hyper-beurocratic Custodians. The Custodians had wiped Soloman's memory before the events of the first film, assigning him Miss Brown as a caretaker to watch over him. In-between the third and fourth films, it's said she's been returned to her home world. The character never appears again in the series.
The Big Finish "Miss Brown" notably meets the Warrior and the Sixth Doctor (both played by Colin Baker) but compares neither to the BBV Stranger character, although this could be explained by the unresolved amnesia storyline from the BBV films. Miss Brown's role in the story implies she is from some alternate reality caused by the Warrior's already alternate timeline Time War.