The most common and most likely answer to this crossword clue is the 9 plus 5 letter word HAMPSTEAD HEATH
I've seen this crossword clue in The Guardian
H
A
M
P
S
T
E
A
D
H
E
A
T
H
More clues leading to the result HAMPSTEAD HEATH
Buddy explains!

The straight part
We know that "Green area’" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case green area’ = hampstead heath.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.