  We show that Siegel modular forms of level \Gamma_0(p^m) are p-adic modular
forms. Moreover we show that derivatives of such Siegel modular forms are
p-adic. Parts of our results are also valid for vector-valued modular forms. In
our approach to p-adic Siegel modular forms we follow Serre closely; his proofs
however do not generalize to the Siegel case or need some modifications.
