The most common and most likely answer to this crossword clue is the 9 letter word NON-DAIRY
I've seen this crossword clue in Financial Times
N
O
N
-
D
A
I
R
Y
More clues leading to the result NON-DAIRY
Buddy explains!

The straight part
We know that "Type of diet" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case type of diet = non-dairy.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.