The most common and most likely answer to this crossword clue is the 7 letter word WHIPPET
I've seen this crossword clue in The Times
W
H
I
P
P
E
T
Buddy explains!

The straight part
We know that "Dog" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case dog = whippet.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 WHIPPET mean?
WHIPPET noun- A dog of a certain breed, similar to a small greyhound, originating in Britain and bred for racing.
- A cartridge of nitrous oxide (laughing gas), used as a recreational inhalant drug.