The most common and most likely answer to this crossword clue is the 3 plus 5 plus 3 letter word ACT ONE’S AGE
I've seen this crossword clue in The Guardian
A
C
T
O
N
E
â
€
™
S
A
G
E
Buddy explains!
The straight part
We know that "Don’t be childish" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case don’t be childish = act one’s age.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.