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From Terraria Wiki
The Adhesive Bandage is a Hardmode accessory that has a 1% / 2% (1/100 / 1/50 ) chance to drop from Rusty Armored Bones, Werewolves and Angler Fish. It grants immunity to any Bleeding debuffs.
It is one of the ingredients required to make the Medicated Bandage, which is a crafting material required to make the Ankh Charm, an accessory further required to make the Ankh Shield.