The most common and most likely answer to this crossword clue is the 9 letter word INELASTIC
I've seen this crossword clue in Financial Times
I
N
E
L
A
S
T
I
C
More clues leading to the result INELASTIC
- Not giving in, ruined castle defends island
- Awfully clean - is it rigid?
- Not giving in, city guards survive one
- Not giving in, knocked back beer with staff endlessly
- Rigid tails, nice when wagging
- Silly line on wine, cold and unyielding
- Sentinel a stickler at heart, refusing to bend
- The Sentinel, a stickler for holding rigid
- A client is fantastically uncompromising
- Plasticine less soft, possibly without resilience
Buddy explains!

The straight part
We know that "Rigid" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case rigid = inelastic.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 INELASTIC mean?
INELASTIC adjective- Lacking elasticity; inflexible, unyielding