The most common and most likely answer to this crossword clue is the 8 letter word ZEALOTRY
I've seen this crossword clue in The Sunday Telegraph
Z
E
A
L
O
T
R
Y
Buddy explains!

The straight part
We know that "Extremism" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case extremism = zealotry.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 ZEALOTRY mean?
ZEALOTRY noun- Something characteristic of a zealot; excessive zeal; fanaticism.