The most common and most likely answer to this crossword clue is the 3 plus 6 letter word FOR TOFFEE
I've seen this crossword clue in The Guardian
F
O
R
T
O
F
F
E
E
More clues leading to the result FOR TOFFEE
Buddy explains!
The straight part
We know that "not one little bit" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case not one little bit = for toffee.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.