The most common and most likely answer to this crossword clue is the 11 letter word JET-SETTING
I've seen this crossword clue in The Times
J
E
T
-
S
E
T
T
I
N
G
Buddy explains!

The straight part
We know that "Gallivanting" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case gallivanting = jet-setting.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 JET-SETTING mean?
JET-SETTING noun- The lifestyle of the jet set; travelling from one fashionable location to another by jet.