The most common and most likely answer to this crossword clue is the 4 plus 4 letter word HELD FIRM
I've seen this crossword clue in The Daily Telegraph
H
E
L
D
F
I
R
M
Buddy explains!
The straight part
We know that "Refused to abandon one’s belief" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case refused to abandon one’s belief = held firm.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.