The most common and most likely answer to this crossword clue is the 7 plus 4 letter word READING ROOM
I've seen this crossword clue in The Daily Telegraph
R
E
A
D
I
N
G
R
O
O
M
Buddy explains!
The straight part
We know that "part of library" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case part of library = reading room.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.