The most common and most likely answer to this crossword clue is the 10 letter word BUTTERMILK
I've seen this crossword clue in The Guardian
B
U
T
T
E
R
M
I
L
K
More clues leading to the result BUTTERMILK
Buddy explains!

The straight part
We know that "Dairy product" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case dairy product = buttermilk.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 BUTTERMILK mean?
BUTTERMILK noun- The liquid left over after producing butter from full cream milk by the churning process, also called traditional buttermilk.
- Cultured buttermilk, a fermented dairy product produced from cow's milk, with a characteristically sour taste.