The most common and most likely answer to this crossword clue is the 6 letter word VESTRY
I've seen this crossword clue in The Times
V
E
S
T
R
Y
More clues leading to the result VESTRY
- Pious figure clothed in most suitable room?
- Saint accommodated in particular part of church
- Where ministers change top lines
- Holy person occupies most room in church
- Good guy in real dressing room
- Where holy man gets appropriate covering
- Man of God confined to actual room in church
- Room in church really enshrining saint
- Way into actual robing room
- Saint occupying extremely religious office
Buddy explains!

The straight part
We know that "part of church" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case part of church = vestry.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 VESTRY mean?
VESTRY noun- A room in a church where the clergy put on their vestments and where these are stored; also used for meetings and classes; a sacristy.
- A committee of parishioners elected to administer the temporal affairs of a parish.
- An assembly of persons who manage parochial affairs; so called because usually held in a vestry.