  Berger and Colmez introduced a theory of families of overconvergent \'etale
(Phi,Gamma)-modules associated to families of p-adic Galois representations
over p-adic Banach algebras. However, in contrast with the classical theory of
(Phi,Gamma)-modules, the functor they obtain is not an equivalence of
categories. In this paper, we prove that when the base is an affinoid space,
every family of (overconvergent) \'etale (Phi,Gamma)-modules can locally be
converted into a family of p-adic representations in a unique manner, providing
the "local" equivalence. There is a global mod p obstruction related to the
moduli of residual representations.
