The most common and most likely answer to this crossword clue is the 6 letter word MUTISM
I've seen this crossword clue in Financial Times
M
U
T
I
S
M
Buddy explains!

The straight part
We know that "Unwillingness to speak" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case unwillingness to speak = mutism.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 MUTISM mean?
MUTISM noun- A psychological disorder in which the sufferer cannot speak in certain situations.