Mechanically speaking, a Divine Smite is a Paladin class ability. Meanwhile, a spell is anything listed in PHB pages 207-211, which shows the spell list for all classes. So even though Divine Smites use spell slots, they are not spells.
Not all spells consume spell slots
From PHB 201:
Some characters and monsters have special abilities that let them cast spells without using spell slots.
Therefore, spells and spell slots do not have to be tied together, though they usually are. Spells can be cast without spell slots. Just because something consumes a spell slot doesn't mean it has to be a spell.
Antimagic Field does affect it
While a Divine Smite is not a spell, it is magical. "Magical things" is a larger set that contains all spells.
Antimagic Field specifies:
Spells and other magical effects, except those created by an artifact or a deity, are suppressed in the sphere and can’t protrude into it.
It specifically states:
Targeted Effects. Spells and other magical effects, such as magic missile and charm person, that target a creature or an object in the sphere have no effect on that target.
As Divine Smite is a magical effect, it is nullified in an Antimagic Field.
It does work in an area of silence
Divine Smite is not a spell and requires no verbal spell components. All it requires is you hit with a weapon and you expend a spell slot.
Starting at 2nd level, when you hit a creature with a melee weapon attack, you can expend one paladin spell slot to deal radiant damage to the target