Drive off in public — go too far? (9)
I believe the answer is:
overshoot
'go too far?' is the definition.
(I've seen this before)
'drive off in public' is the wordplay.
'drive off' becomes 'shoo' (shooing is a kind of driving off).
'in' is an insertion indicator.
'public' becomes 'overt' ('overt' is associated in meaning with 'public').
'shoo' going into 'overt' is 'OVERSHOOT'.
(Other definitions for overshoot that I've seen before include "Maybe miss the green" , "Taxi too much, perhaps" , "Fly beyond" , "Go beyond the mark" , "Go past unintentionally" .)