The most common and most likely answer to this crossword clue is the 10 letter word CATNAPPING
I've seen this crossword clue in The Guardian
C
A
T
N
A
P
P
I
N
G
Buddy explains!
The straight part
We know that "that’s briefly asleep" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case that’s briefly asleep = catnapping.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 CATNAPPING mean?
CATNAPPING verb- To take a catnap, to take a short sleep or nap.
- To kidnap a cat.
- The kidnapping or theft of a cat.