The most common and most likely answer to this crossword clue is the 4 letter word MAMA
M
A
M
A
Buddy explains!
The straight part
We know that "mature lady (in US)" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case mature lady (in us) = mama.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 MAMA mean?
MAMA noun- (hypocoristic, usually childish) Mother, female parent.