Polymorph states (in part)

The target’s game statistics, including mental ability scores, are replaced by the statistics of the chosen beast. It retains its alignment and personality.

The creature is limited in actions it can perform by the nature of its new form, and it can’t speak, cast spells, or take any other action that requires hands, or speech.

So, can a polymorphed Paladin use their Holy Smite class feature? It expends a spell slot, but it is not a spell, and as far as we can tell does not require hands or speech.