  Following Apery's proof of the irrationality of zeta(3), Beukers found an
elegant reinterpretation of Apery's arguments using modular forms. We show how
Beukers arguments can be adapted to a p-adic setting. In this context, certain
functional equations arising from Eichler integrals are replaced by the notion
of overconvergent p-adic modular forms, and the periods themselves arise not as
coefficients of period polynomials but as constant terms of p-adic Eisenstein
series. We prove that the analogue of zeta(3) is irrational for p = 2 and 3, as
well as the 2-adic analogue of Catalan's constant.
