The most common and most likely answer to this crossword clue is the 6 letter word PLEURA
P
L
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U
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A
More clues leading to the result PLEURA
Buddy explains!

The straight part
We know that "lung membrane" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case lung membrane = pleura.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 PLEURA mean?
PLEURA noun- The smooth serous membrane which closely covers the lungs and the adjacent surfaces of the thorax; the pleural membrane.
- A lateral sclerite of a thoracic segment of an arthropod between the tergum and the sternum.