The most common and most likely answer to this crossword clue is the 9 letter word PRESERVER
I've seen this crossword clue in Financial Times
P
R
E
S
E
R
V
E
R
Buddy explains!

The straight part
We know that "National Trust possibly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case national trust possibly = preserver.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 PRESERVER mean?
PRESERVER noun- One who preserves.
- A life preserver.
- A person who refinishes furniture.
- A person who prepares preserves of fruit or preserved meats.