Can you react to a unit dying?

Yes — with an important distinction. You can't react to the death event itself, but you can react to whatever is on the Chain that leads to it.

What you CAN react to:

• The effect that kills the unit — a kill/damage spell, an activated ability, or a trigger like Temporary — while it's still on the Chain (Rule 359.3, FAQ #2751).
• Triggered abilities that fire because of the death, like Deathknell or "When I die…" — those go on the Chain as pending items and create a reaction window (Rules 323.4, 428.1.a.1.b).

What you CANNOT react to:

• The death itself. When a kill instruction resolves or lethal damage is checked in a Cleanup, the unit dies immediately with no priority granted — you can't slip a save in between (FAQ #2971, FAQ #11388).

So: react to the cause, or to the death-trigger — but not to the moment of death.