The most common and most likely answer to this crossword clue is the 12 letter word NON-RESIDENT
I've seen this crossword clue in The Times
N
O
N
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R
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I
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T
Buddy explains!
The straight part
We know that "one that can’t stay" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case one that can’t stay = non-resident.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 NON-RESIDENT mean?
NON-RESIDENT noun- A person living in a country who is no legal permanent resident.
- One who is not a resident; an alien; a foreigner.