The most common and most likely answer to this crossword clue is the 4 letter word FAWN
I've seen this crossword clue in The Times
F
A
W
N
More clues leading to the result FAWN
Buddy explains!

The straight part
We know that "Behave like obsequious dog/other little animal" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case behave like obsequious dog/other little animal = fawn.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 FAWN mean?
FAWN noun- A young deer.
- A pale brown colour tinted with yellow, like that of a fawn.
- The young of an animal; a whelp.
- To give birth to a fawn.
- Of the fawn colour.
- A servile cringe or bow.
- Base flattery.
- To exhibit affection or attempt to please.
- To seek favour by flattery and obsequious behaviour (with on or upon).
- (of a dog) To show devotion or submissiveness by wagging its tail, nuzzling, licking, etc.