The most common and most likely answer to this crossword clue is the 6 letter word NEVER!
I've seen this crossword clue in The Guardian
N
E
V
E
R
!
More clues leading to the result NEVER!
Buddy explains!

The straight part
We know that "I don’t believe it!" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case i don’t believe it! = never!.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 NEVER! mean?
NEVER! adverb- At no time; on no occasion; in no circumstance.
- Not at any other time; not on any other occasion; not previously.
- Negative particle (used to negate verbs in the simple past tense; also used absolutely).