The most common and most likely answer to this crossword clue is the 12 letter word EXTERMINATED
I've seen this crossword clue in Financial Times
E
X
T
E
R
M
I
N
A
T
E
D
Buddy explains!

The straight part
We know that "Finished off" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case finished off = exterminated.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 EXTERMINATED mean?
EXTERMINATED verb- To kill all of (a population of pests or undesirables), usually intentionally.
- To bring a definite end to; finish completely.