The most common and most likely answer to this crossword clue is the 8 letter word DEPARTED
I've seen this crossword clue in The Guardian
D
E
P
A
R
T
E
D
More clues leading to the result DEPARTED
Buddy explains!

The straight part
We know that "quit" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case quit = departed.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 DEPARTED mean?
DEPARTED verb- To leave.
- To set out on a journey.
- To die.
- To disappear, vanish; to cease to exist.
- To deviate (from), be different (from), fail to conform.
- To go away from; to leave.
- To divide up; to distribute, share.
- To separate, part.
- A dead person or persons.
- Dead