The most common and most likely answer to this crossword clue is the 11 letter word CHAUFFEURED
I've seen this crossword clue in The Times
C
H
A
U
F
F
E
U
R
E
D
Buddy explains!

The straight part
We know that "Drove" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case drove = chauffeured.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does CHAUFFEURED mean?
CHAUFFEURED verb- To be, or act as, a chauffeur (driver of a motor car).
- To transport (someone) in a motor vehicle.