The 53-second clip and two pictures – which were uploaded on American messaging app Tapatalk – both feature actress Jodie Whittaker, the thirteenth Doctor in the sci-fi classic.
In response, the BBC has asked the California Federal Court to force the company, which is based in California, to hand over records which could help identify the person or people responsible for the leak.
The company itself is not accused of wrongdoing and the terms and conditions of their site make clear that uploading copyrighted material is not allowed.
A spokesman for the BBC said: “BBC Studios takes the theft and illegal distribution of our content extremely seriously.
“We will strive to protect our programme-makers, audiences and licence fee payers from any breaches of security – ensuring Doctor Who fans enjoy the final and fully completed version of the episode when it premieres.”
The unauthorised clip is understood to have been an unfinished piece of footage, without finessed lighting or music, which would be added in post-production.
It is also alleged that a discussion about the short scene took place on a second online platform – Discord, which is mainly used by gaming communities – almost a week before it was shared on Tapatalk.
Some Doctor Who fans have speculated that the footage may have been intended for use at the San Diego Comic-Con, which will be attended by some of the show’s cast and crew, including Jodie Whittaker, later this month.
After a brief appearance at the end of the 2017 Doctor Who Christmas special, this series will be Whittaker’s first full-length appearance as the Doctor, with Bradley Walsh taking on the role of her companion.
The eleventh series of Doctor Who will air on the BBC in the autumn, kicking off with a one-hour special.