Why does misinformation persist, and how does it distort the long-run beliefs and actions of rational agents? Suppose receivers see an infinite stream of messages from a sender of unknown type who observes private signals about an unknown state of the world. We characterize the conditions for “doublespeak” equilibria where one sender type repeatedly reveals each private signal truthfully but another sender type repeatedly fabricates false values of her private signals. Receivers only partially learn the true state in the long run irrespective of the true sender type, resulting in long-run disagreement and ex post incorrect actions by some receivers. Equilibrium fact-checking by receivers does not induce more truth-telling among sender types but reputational concerns can. Our results cast doubt on the presumption that rational agents can pierce through persistent extreme lies in the long run and highlight the deleterious effects of such lies for receiver welfare.