The most common and most likely answer to this crossword clue is the 7 letter word EXITING
I've seen this crossword clue in The Daily Telegraph
E
X
I
T
I
N
G
More clues leading to the result EXITING
Buddy explains!

The straight part
We know that "Finishing playing one’s part" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case finishing playing one’s part = exiting.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 EXITING mean?
EXITING verb- To go out or go away from a place or situation; to depart, to leave.
- To depart from life; to die.
- To end or terminate (a program, subroutine, etc.)
- (originally United States) To depart from or leave (a place or situation).
- The act of one who exits; a departure.