The most common and most likely answer to this crossword clue is the 2 plus 1 plus 5 plus 5 letter word IN A CLEFT STICK
I've seen this crossword clue in The Daily Telegraph
I
N
A
C
L
E
F
T
S
T
I
C
K
Buddy explains!
The straight part
We know that "Facing a dilemma" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case facing a dilemma = in a cleft stick.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.