In Persona, the Identity Provider is not involved in each login, it just signs a temporary certificate which can be re-used by the browser, so as long as the downtime is under a few hours, the user shouldn't have much of a problem.
And if the Identity Provider's gone for a prolonged period now you've lost your identity with (almost) no means of recovery. Mostly, because, while you might believed the contrary, you didn't ever own your "own" identity in this scheme.