The most common and most likely answer to this crossword clue is the 4 plus 2 plus 6 letter word KEEP ON TRYING
I've seen this crossword clue in The Daily Telegraph
K
E
E
P
O
N
T
R
Y
I
N
G
Buddy explains!
The straight part
We know that "Don’t give up" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case don’t give up = keep on trying.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.