The most common and most likely answer to this crossword clue is the 4 letter word LIAR
I've seen this crossword clue in The Guardian
L
I
A
R
More clues leading to the result LIAR
- Story-teller in bar getting upset
- Trouble over Republican - one committing perjury
- On return, trains to become make-up artist?
- Return of transport system? He’s not to be believed
- Story-teller finds a means of travelling back
- Revolving bar, one serving pork pies
- Diamonds missed by detection system, one giving false response
- Method of travel on rise? Don't believe him!
- Story teller in trouble over reading?
- Romancer raising the bar
Buddy explains!

The straight part
We know that "False witness" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case false witness = liar.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 LIAR mean?
LIAR noun- One who tells lies.
- A swabber responsible for cleaning the outside parts of the ship rather than the cabins, a role traditionally assigned to a person caught telling a lie the previous week.