The most common and most likely answer to this crossword clue is the 3 letter word FIB
I've seen this crossword clue in Independent
F
I
B
Buddy explains!

The straight part
We know that "false claim" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case false claim = fib.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 FIB mean?
FIB noun- A lie, especially one that is more or less inconsequential.
- A liar.
- To lie, especially more or less inconsequentially.
- The fibula.
- (thieves cant) To punch, especially a series of punches in rapid succession; to beat; to hit; to strike.
- A kind of experimental poem where the number of syllables in each line is the next succeeding Fibonacci number.