The most common and most likely answer to this crossword clue is the 13 letter word ASSASSINATING
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The straight part
We know that "Killing" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case killing = assassinating.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.
Crossword Answer definitions
ASSASSINATING verb- To murder someone, especially an important person, by a sudden or obscure attack, especially for ideological or political reasons.
- To harm, ruin, or defame severely or destroy by treachery, slander, libel, or obscure attack.