When The Doctor, Bill and Nardole land on another planet, the windows of the TARDIS are seen completely white (which means the lights are on). When the camera gets near to the TARDIS, the lights are off and in the distant shots the lights are on again.
In a close-up shot of Heather during the "You're dead!" exchange, the edge of her contact lens can briefly be seen.
Just before the Doctor tries to erase Bill's memory, he says he was late because a student getting sick set off the vault's security alarm. Since the way to the vault can only be accessed by the Doctor, Nardole, an (accidentally) Bill, there's no way a student could have gotten close enough to set off the alarm.
The Doctor says that "nobody's from space," and that he's, "from planet like everybody else". However he forgot about space stations and multi-generational ships, two things that he's encountered previously.