The most common and most likely answer to this crossword clue is the 5 plus 3 plus 3 letter word NEVER SAY DIE
I've seen this crossword clue in The Daily Telegraph
N
E
V
E
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S
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More clues leading to the result NEVER SAY DIE
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 = never say die.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 NEVER SAY DIE mean?
NEVER SAY DIE adjective- That does not give up in despair; indomitable.