The most common and most likely answer to this crossword clue is the 16 letter word COLD-SHOULDERING
I've seen this crossword clue in The Guardian
C
O
L
D
-
S
H
O
U
L
D
E
R
I
N
G
Buddy explains!

The straight part
We know that "Delivering rebuff" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case delivering rebuff = cold-shouldering.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 COLD-SHOULDERING mean?
COLD-SHOULDERING verb- To disrespect someone, especially by ignoring them.