The most common and most likely answer to this crossword clue is the 6 letter word JOSHUA
I've seen this crossword clue in The Guardian
J
O
S
H
U
A
More clues leading to the result JOSHUA
- Susan regularly found behind joke book
- Biblical character having joke with Judah, oddly ignored
- Rib leading to human, oddly; ousted in Bible book
- Banter to lift a high-class fighter
- Tease superior with a religious book
- Boxer has to tease U3A, if missing tweet here
- Book a top fighter
- Joseph’s with Hugh reportedly taking a book
- Hoax leads to unfortunate arrest — I’m before Judges
- Ridicule you in text with answer in book
Buddy explains!

The straight part
We know that "fighter" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case fighter = joshua.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.