The Editor in Chief of Taxation magazine speculates on this question in the 14 Jan issue.
He wonders how she would fare under the statutory residence test as she seems to have spent a lot of her time in the UK.
Perhaps in the next series her deadliest enemy could turn out not to be a dalek but someone from HMRC's anti-avoidance unit. 'Dr Who and the follower notice'. You heard it here first.
The answer to the question though is clearly: No, Dr Who is not liable to pay UK income tax.
The character has no permanent home here, was not born here and earns no money here. Plus, and we should keep this in mind, she's a fictional character allegedly from another planet. HMRC have no taxing powers over such people, real or not ;-)