The most common and most likely answer to this crossword clue is the 4 plus 1 plus 4 plus 2 plus 4 plus 3 letter word HAVE A MIND OF ONES OWN
I've seen this crossword clue in The Times
H
A
V
E
A
M
I
N
D
O
F
O
N
E
S
O
W
N
Buddy explains!
The straight part
We know that "Act independently" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case act independently = have a mind of ones own.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.