The most common and most likely answer to this crossword clue is the 9 letter word DEFERMENT
I've seen this crossword clue in The Guardian
D
E
F
E
R
M
E
N
T
More clues leading to the result DEFERMENT
- Seemingly take the excitement out of stay
- Furore following elevated journalist’s stay
- Stay in midst of students’ agitation
- Putting off the calming of excitement?
- To reduce agitation apparently is putting off
- Delay due to unrest by lower socioeconomic groups
- Delay of French Revolution’s effect
- Fed up at the Queen’s soldiers’ ultimate transport delay
- Delay of the French revolution
Buddy explains!

The straight part
We know that "Stay" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case stay = deferment.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 DEFERMENT mean?
DEFERMENT noun- An act or instance of deferring or putting off.
- Officially sanctioned postponement of compulsory military service.