The most common and most likely answer to this crossword clue is the 6 letter word BALLED
I've seen this crossword clue in The Times
B
A
L
L
E
D
Buddy explains!

The straight part
We know that "curled up" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case curled up = balled.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 BALLED mean?
BALLED verb- To form or wind into a ball.
- To heat in a furnace and form into balls for rolling.
- To have sexual intercourse with.
- To gather balls which cling to the feet, as of damp snow or clay; to gather into balls.
- (usually in present participle) To be hip or cool.
- To play basketball.
- To punish by affixing a ball and chain
- (in combination) Having a specified kind or number of balls.