Continuity mistake: When the Captain's arrival at the South Pole is first shown, walking up while Twelve is explaining to One that the frozen snow means something is very wrong with time and asking if either of them is a doctor, to which Twelve responds, "You trying to be funny?", the two Doctors are standing right next to each other. When the Captain's arrival is shown from his perspective, however, the Doctors are standing much further apart when Twelve says the line. (00:09:00)
Other mistake: After the Doctor regenerates, there's a close-up of the Thirteenth Doctor's eye with the console room reflected in it. The reflection is the same orientation as a shot from behind her immediately prior, which should not be the case. Take note of the positions of the door and time rotor in both shots. (01:00:45)
Deliberate mistake: The Doctor (Jodie Whittaker) is wearing the same clothes that the previous incarnation (Peter Capaldi) wore after regenerating. Peter Capaldi is 6' and Jodie Whittaker is 5'6", but the clothes fit perfectly. She even says her legs used to be longer, and yet the trousers aren't flapping around her feet.
Factual error: Kandoka labour laws require companies like Kerblam! to have a minimum of 10% organic staff. The villain claims that, as a result, only 10% of humans are employed, at all. These are two completely different quantities: if 10% of the cars in one parking lot are blue, this doesn't indicate that 10% of all cars are painted blue, but that's effectively what the episode thinks. The fact that no-one else, including the Doctor, calls him on this further magnifies the mathematical error.