The most common and most likely answer to this crossword clue is the 9 letter word ABNEGATES
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More clues leading to the result ABNEGATES
Buddy explains!

The straight part
We know that "gives up" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case gives up = abnegates.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 ABNEGATES mean?
ABNEGATES verb- To deny (oneself something); to renounce or give up (a right, a power, a claim, a privilege, a convenience).
- To relinquish; to surrender; to abjure.