In a 1987 letter to Tate, Serre showed that the Hecke eigensystems appearing in mod p modular forms are the same as those appearing in mod p functions on a finite double coset constructed from the quaternion algebra ramified at p and infinity. At the end of the letter, he asked whether there might be a similar relation between p-adic modular forms and p-adic functions on the quaternion algebra. We show the answer is yes: the completed Hecke algebra of p-adic modular forms is the same as the completed Hecke algebra of naive p-adic automorphic functions on the quaternion algebra. The resulting p-adic Jacquet-Langlands correspondence is richer than the classical Jacquet-Langlands correspondence -- for example, Ramanujan's delta function, which is invisible to the classical correspondence, appears. The proof is a lifting of Serre's geometric argument from characteristic p to characteristic zero; the quaternionic double coset is realized as a fiber of the Hodge-Tate period map, and eigensystems are extended off of the fiber using a variant of Scholze's fake Hasse invariants.