Can someone tell me if the following is a compound sentence please?

"They finally leave but still stay close to their mothers"

Thank you!

Your sentence has a main clause and another clause with the coordinating conjunction 'but'. So far, so good.

A compound sentence contains two or more INDEPENDENT clauses joined together with a coordinating conjunction.

An independent clause is a clause that contains three things:

1. A subject
Your sentence has no subject

2. A verb / an action
The verb is 'stay'

3. A complete thought (there are no questions as to meaning at the end of the sentence)
"Still stay close to their mother" is not a complete thought. The first question a reader would ask is 'who stay close to their mother'? a woman's children? puppies? ducklings?

So, no, it isn't.

Here's a grammar lesson on compound sentences.