The most common and most likely answer to this crossword clue is the 9 plus 6 letter word SURROGATE MOTHER
I've seen this crossword clue in The Guardian
S
U
R
R
O
G
A
T
E
M
O
T
H
E
R
More clues leading to the result SURROGATE MOTHER
Buddy explains!
The straight part
We know that "Carrier of baby" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case carrier of baby = surrogate mother.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.