palm has 8 distinct synonyms spread across 4 of its 5 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.
a linear unit based on the length or width of the human hand
any plant of the family Palmae having an unbranched trunk crowned by large pinnate or palmate leaves
an award for winning a championship or commemorating some other event
Synonyms, antonyms and the broader and narrower relations for palm come from Princeton University’s WordNet. Full sources and licences.