Icefall Regent’s triggered ability can target any creature controlled by an opponent, including one that is already tapped. In that case, the creature stays tapped and won’t untap during its controller’s untap step.

If another player gains control of Icefall Regent, the creature will no longer be affected by the ability stopping it from untapping, even if you later regain control of Icefall Regent.

Icefall Regent’s last ability affects all spells cast by your opponents that target it, including Aura spells and spells with multiple targets. It doesn’t affect abilities.

The ability stopping the creature from untapping will continue to apply to it even if the creature changes controllers.