The most common and most likely answer to this crossword clue is the 12 letter word CAPITULATION
I've seen this crossword clue in The Times
C
A
P
I
T
U
L
A
T
I
O
N
Buddy explains!

The straight part
We know that "giving up" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case giving up = capitulation.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 CAPITULATION mean?
CAPITULATION noun- A reducing to heads or articles; a formal agreement.
- The act of capitulating or surrendering to an enemy upon stipulated terms; the act of ceasing to resist an opponent or an unwelcome demand.
- The instrument containing the terms of an agreement or surrender.
- An enumeration of the main parts of a subject.