  Motivated by the study of algebraic classes in mixed characteristic we define
a countable subalgebra of $\bar{\mathbb{Q}}_p$ which we call the algebra of
Andr\'e's $p$-adic periods. We construct a tannakian framework to study these
periods. In particular, we bound their transcendence degree and formulate the
analog of the Grothendieck period conjecture. We exhibit several examples where
special values of classical $p$-adic functions appear as Andr\'e's $p$-adic
periods and we relate these new conjectures to some classical problems on
algebraic classes.
