Banishing Light’s ability causes a zone change with a duration, a new style of ability that’s somewhat reminiscent of older cards like Oblivion Ring. However, unlike Oblivion Ring, cards like Banishing Light have a single ability that creates two one-shot effects: one that exiles the permanent when the ability resolves, and another that returns the exiled card to the battlefield immediately after Banishing Light leaves the battlefield.

If Banishing Light leaves the battlefield before its enters-the-battlefield ability resolves, the target nonland permanent won’t be exiled.

Auras attached to the exiled permanent will be put into their owners’ graveyards. Equipment attached to the exiled permanent will become unattached and remain on the battlefield. Any counters on the exiled permanent will cease to exist.

If a token is exiled, it ceases to exist. It won’t be returned to the battlefield.

The exiled card returns to the battlefield immediately after Banishing Light leaves the battlefield. Nothing happens between the two events, including state-based actions. Banishing Light and the returned card aren’t on the battlefield at the same time.

In a multiplayer game, if Banishing Light’s owner leaves the game, the exiled card will return to the battlefield. Because the one-shot effect that returns the card isn’t an ability that goes on the stack, it won’t cease to exist along with the leaving player’s spells and abilities on the stack.