She didn't even change her costume.

Santa Claus, Father Christmas, Saint Nicholas, or his lesser known title: The Rani, is clearly The Rani

Tardis Infiltration Edit

Santa Claus in "Death in Heaven" manages not only to infiltrate The Doctor's Tardis, but also to interrupt the very credits of the show itself. Such power can clearly only belong to a Time Lord

Present Delivery Edit

Santa being a Time Lord would easily explain his ability to deliver hundreds of millions of presents within a 24 hour period, his ability to store said presents, and his ability to enter homes undetected. No other species is advanced enough to pull off such a feat.

Other Evidence Edit

Santa and The Rani clearly have extremely similar costumes, and they both employ man servants to do their more menial work. Perhaps most telling of all- The Rani hates both Christmas and children, as such none would ever suspect her of secretly being Santa Claus.