The most common and most likely answer to this crossword clue is the 7 letter word NUMBING
N
U
M
B
I
N
G
Buddy explains!

The straight part
We know that "Killing feeling" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case killing feeling = numbing.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
NUMBING verb- To cause to become numb (physically or emotionally).
- To cause (a feeling) to be less intense.
- To cause (the mind, faculties, etc.) to be less acute.
- To become numb (especially physically).
- An act by which something is numbed.