give up has 31 distinct synonyms spread across 12 of its 12 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
give up in the face of defeat of lacking hope; admit defeat
“In the second round, the challenger gave up”
put an end to a state or an activity
“Quit teasing your little brother”
give up what is not strictly needed
“he asked if they could spare one of their horses to speed his journey”
part with a possession or right
“I am relinquishing my bedroom to the long-term house guest”
leave (a job, post, or position) voluntarily
“She vacated the position when she got pregnant”
give up or agree to forgo to the power or possession of another
“The last Taleban fighters finally surrendered”
Synonyms, antonyms and the broader and narrower relations for give up come from Princeton University’s WordNet. Full sources and licences.