The most common and most likely answer to this crossword clue is the 10 letter word ENGLISHMAN
I've seen this crossword clue in The Daily Telegraph
E
N
G
L
I
S
H
M
A
N
Buddy explains!
The straight part
We know that "Mad dog’s companion in The Sun?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case mad dog’s companion in the sun? = englishman.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.