The Polymorph spell 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 Divine 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.
We know (based on the 10th level Transmutation Wizard feature) that a wizard is supposed to be able to polymorph herself, and we can assume that she is also supposed to be able to maintain concentration needed to stay in the new form (otherwise it would be a pretty useless ability). This is (as far as I can tell) not something any of the beasts eligible as polymorph targets can normally do. So it is clear that the polymorphed creature maintains some abilities from her old form. The question is which ones.
The Druid’s Wild Shape ability explicitly says you can use class features, while Polymorph does not, suggesting that you can’t use class features while polymorphed. However, Wild Shape also explicitly says that you can continue to concentrate on a spell (see below, emphasis mine), and Polymorph does not, yet we believe that a polymorphed creature can continue to concentrate on the polymorph spell. So the fact that Wild Shape explicitly allows something and Polymorph is not explicit about it does not necessarily mean that the polymorphed creature cannot do those things.
From Wild Shape rules:
You can’t cast spells, and your ability to speak or take any action that requires hands is limited to the capabilities of your beast form. Transforming doesn’t break your concentration on a spell you’ve already cast, however, or prevent you from taking actions that are part of a spell, such as call lightning, that you’ve already cast.
So, since Polymorph is silent on the question of whether or not class abilities can be used, and states only that the polymorphed creature cannot perform actions that require hands or speech, how can we best determine whether (RAI) polymorphed creatures maintain any of their old abilities beyond spell concentration?