From the Super Mario Wiki, the Mario encyclopedia
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Paper Mario series
In Paper Mario, if Mario is Frozen, he will not be able to move for several turns. It is indicated by a very large irregular chunk of ice surrounding Mario. The Crystal King can inflict this status on Mario with his Ice Bolt attack.
Paper Mario: The Thousand-Year Door
The status ailment returned in Paper Mario: The Thousand-Year Door and serves much the same purpose, with one key difference. When Mario, his partner, or any enemy's Frozen status expires, that character will take 1 HP of damage when the ice breaks. It is indicated by a hexagonal chunk of ice only as big as the character surrounding them. In addition, if any attack that can induce a Burn is used on a Frozen character, it will negate both status ailments.
Super Paper Mario
The status effects returns in the Super Paper Mario, several items including Fruit Parfait and Snow Bunny will freeze enemies if used in battle. The Ice Cherbil is one of the enemies that can freeze the heroes. Several bosses also have this ability, including Bonechill, whose freeze is more powerful than any other found in game. Mario does no longer take damage by waiting, however getting touched by enemies will make Mario lose HP. The player can only break free if they shake the Wii Remote.
Paper Mario: Sticker Star
Mario + Rabbids Kingdom Battle
The status effect makes an appearance in Mario + Rabbids Kingdom Battle, being named Freeze. Instead of immobilizing characters completely, inflicted characters cannot use any of their techniques for one turn. While inflicted characters can wait until the next turn, as with other prevention status ailments, they can also clear it by having an ally perform a team jump with the inflicted character.