Yes. Such as your body, which you hold private from everyone else in using it. That private, or exclusive use, property changes hands does not mean that it's anything other than private. If you homestead a piece of land, that land is private. You exercise exclusive control over it. This is the case without a state.
A piece of land cannot simultaneously be under the control of two competing individuals simultaneously. Coincidentally, that's where the tragedy of the commons arises from.