Been thinking about this one for ages now - the small animal combined with the learner might suggest the word: GOLFER - GOPHER with 'L' a driver is used in golf, etc etc. I'm having a problem, however, with the phrase 'run over' . If I'm correct perhaps you can explain the last bit of hidden text.

[Chalky] I believe you've got it: Small animal = GOPHER. Small animal said = GOFER. "Run over by learner" means insert an L, hence GOLFER, who drives from the tee.

[Raak] Thanks. 'run over by' as an insertion indicator is new to me, but the rest is much as I thought. Have we solved all the outstanding ones now? If so, before the serious stuff begins again, here's a little light-hearted diversion:
Be alert when one's aurally deviating? (5, 2, 4, 4)

[Chalky] Well I reckon that must be Prick up ones ears (or "your" for "ones" - but "ones" is commoner in Xwd answers).
Still my Presume the leader is back in place (8) to do. rab may well complain that the penultimate word is unnecessary and confusing, so you can ignore it if you like.

[Chalky] Have a coconut! My understanding about the use of terms like run over is that it's ok if it could be taken in the intended crossword sense without too much difficulty. This might be on the edge of fair play, but lenience may be granted as it carries on the English language sense of the clue, and strengthens the misdirection of "driver" as a controller of automotive transport. Well, anyway.