The most common and most likely answer to this crossword clue is the 11 letter word SURRENDERED
I've seen this crossword clue in The Guardian
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More clues leading to the result SURRENDERED
Buddy explains!

The straight part
We know that "gave up/gave up" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case gave up/gave up = surrendered.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 SURRENDERED mean?
SURRENDERED verb- To give up into the power, control, or possession of another.
- (by extension) To yield (a town, a fortification, etc.) to an enemy.
- To give oneself up into the power of another, especially as a prisoner; to submit or give in.
- To give up possession of; to yield; to resign.
- To yield (oneself) to an influence, emotion, passion, etc.
- To abandon (one's hand of cards) and recover half of the initial bet.
- For a policyholder, to voluntarily terminate an insurance contract before the end of its term, usually with the expectation of receiving a surrender value.