depends on the definition you're using, the primary one simply states that a Persona is simply how we present ourselves to society, whether its our true personality or not. As such it wouldn't be possible to not have a persona, unless we simply never have any interactions with other human beings

If you're talking about in terms of Jungian Psychology uhh... No. No it's not. As long as you interact with other people you have a persona. To not have one would be to literally not have a personality in any way, shape, or form, which isn't possible without being a vegetable.

Even staying within the game's canon it's kinda impossible unless you literally have no sense of self.