The most common and most likely answer to this crossword clue is the 4 plus 7 letter word TREE SURGEON
I've seen this crossword clue in Financial Times
T
R
E
E
S
U
R
G
E
O
N
More clues leading to the result TREE SURGEON
- One might slice cherries and our greenest bananas
- Climber dealing with a broken limb?
- Might I grab your lime and cut it?
- Bad steer, to incite one who may go out on a limb
- Troy’s first prophet turned up to encourage woodworker
- Press on after bum steer from woodworker
- Doctor on a plane, perhaps?
- He might take blade to Elm Street, without bounds he operates?
- One may treat elder son with greed, mostly – true, sadly
- One who’s trained to work on a plane?
Buddy explains!

The straight part
We know that "woodworker" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case woodworker = tree surgeon.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.