The most common and most likely answer to this crossword clue is the 9 letter word UNBENDING
I've seen this crossword clue in The Daily Telegraph
U
N
B
E
N
D
I
N
G
Buddy explains!

The straight part
We know that "Inflexible" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case inflexible = unbending.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 UNBENDING mean?
UNBENDING adjective- Inflexible and not yielding
- Very reserved, aloof and asocial
- To remove a bend so as to make, or allow to become, straight
- To release (a load) from a strain or from exertion; to set at ease for a time; to relax.
- To unfasten sails from the spars or stays to which are attached for use.
- To cast loose or untie
- To cease to be bent; to become straight.
- To relax in exertion, attention, severity, or the like
- To enjoy oneself; to become affable and free from formality
- The act of one who unbends.