I'm sure someone else has already observed this, so my apologies if I'm duplicating somebody's previous posting.
Anyway... Clara spread herself through the Doctor's timestream to save him. To nudge him in the right direction.
In one of those, she shows up on Gallifrey to make sure the Doctor takes a specific TARDIS. The obvious thought is that Clara, in that version of herself, was a Gallifreyan, if not actually a Time Lady.
However... in The Doctor's Wife, Idris says that she (the TARDIS) chose him.
So, is Clara the TARDIS?
It probably didn't mean anything, but every time I think of this, I think of River's "ghost" asking the Doctor, "If she's dead, why am I still here?" and then shushing him with "spoilers" when he tries to probe the topic...
... but that's a good question. How did Clara survive being disseminated through the Doctor's timeline? Supposedly, that's not possible.