The most common and most likely answer to this crossword clue is the 10 letter word AUTONOMOUS
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Buddy explains!
The straight part
We know that "Self-driving" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case self-driving = autonomous.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 AUTONOMOUS mean?
AUTONOMOUS adjective- Self-governing. Intelligent, sentient, self-aware, thinking, feeling, governing independently.
- Acting on one's own or independently; of a child, acting without being governed by parental or guardian rules.
- (Celtic linguistics, of a verb form) Used with no subject, indicating an unknown or unspecified agent; used in similar situations as the passive in English (the difference being that the theme in the English passive construction is the subject, while in the Celtic autonomous construction the theme is the object and there is no subject).