The most common and most likely answer to this crossword clue is the 7 letter word FUNDIES
I've seen this crossword clue in The Times
F
U
N
D
I
E
S
Buddy explains!

The straight part
We know that "some strict believers" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case some strict believers = fundies.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 FUNDIES mean?
FUNDIES noun- A fundamentalist Christian.
- (by extension) A religious fundamentalist of any faith.