The most common and most likely answer to this crossword clue is the 4 letter word CERO
C
E
R
O
More clues leading to the result CERO
Buddy explains!

The straight part
We know that "Big fish" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case big fish = cero.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 CERO mean?
CERO noun- A large scombroid food fish (Scomberomorus regalis) found chiefly in the West Indies.