The most common and most likely answer to this crossword clue is the 10 letter word FEATHERBED
F
E
A
T
H
E
R
B
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D
Buddy explains!

The straight part
We know that "baby" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case baby = featherbed.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 FEATHERBED mean?
FEATHERBED noun- A mattress stuffed with feathers.
- (Dartmoor) A bog covered by a layer of moss, presenting a hazard to walkers.
- To treat someone with excessive indulgence; to pamper, cosset or mollycoddle.