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We introduce a hitherto unexplored form of quantum nonlocality, termed local subset unidentifiability, that arises from the limitation of spatially separated parties to perfectly identify a subset of mutually orthogonal multipartite quantum states, randomly chosen from a larger known set, using Local Operations and Classical Communication (LOCC). We show that this nonlocality is stronger than other existing forms of quantum nonlocality, such as local indistinguishability and local unmarkability. If more than one multipartite states from a locally indistinguishable set are distributed between spatially separated parties in a sequentially ordered fashion, then they may or may not mark which state is which using LOCC. However, we show that even when the parties cannot mark the states, they may still locally identify the particular states given to them, though not their order -- i.e., they can identify the elements of the given subset of states. Then we prove the existence of such subsets that are not even locally identifiable, thereby manifesting a stronger nonlocality. We also present the genuine version of this nonlocality -- genuine subset unidentifiability -- where the provided subset remains unidentifiable unless all the parties come together in a common location and perform global measurements. We anticipate potential applications of this nonlocality for future quantum technologies. We discuss one such application in a certain secret password distribution protocol, where this nonlocality outperforms its predecessors as a resource.