The most common and most likely answer to this crossword clue is the 9 plus 5 letter word THREE-DAY EVENT
I've seen this crossword clue in The Guardian
T
H
R
E
E
-
D
A
Y
E
V
E
N
T
Buddy explains!
The straight part
We know that "Badminton" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case badminton = three-day event.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.