Smite

Death-saving effects can prevent the banish, because the unit's owner decides which replacement effect applies first and can apply their protective effect before Smite's.

Smite's "if it would die this turn, banish it instead" is a replacement effect.^[369.1] So are each of the three protective effects, which all replace the event of a unit dying. When multiple replacement effects apply to the same event, the owner of the affected unit decides the order they apply.^[372]

If the unit's owner applies their protective effect first, the death event is fully replaced — the unit is healed, exhausted, and recalled, or an equipment dies in its place. A replacement effect can only be applied once to an event, and to any events that replace it;^[370.2] once the death event has been consumed, Smite's replacement effect has no qualifying event left to intercept, and the unit is not banished.

Conversely, if Smite's effect is applied first, the unit is banished. The protective effects each replace a "die" event — since banishment is not a subset of dying,^[427.2.a] none of them can chain onto the resulting "banish" event. That said, a unit's owner would never choose this order if their intent is to save it, so Smite effectively cannot banish a unit that has a death-saving effect protecting it.