To a torus action on a complex vector space, Gelfand, Kapranov and
Zelevinsky introduce a system of differential equations, called the
GKZ hypergeometric system. Its solutions are GKZ hypergeometric
functions. We study the $p$-adic counterpart of the GKZ
hypergeometric system. In the language of dagger spaces introduced by Grosse-Kl\"onne, the $p$-adic GKZ hypergeometric
complex is a twisted relative de Rham complex of meromorphic
differential forms with logarithmic poles for an affinoid toric dagger space over the dagger unit polydisc. It is a complex
of ${\mathcal O}^\dagger$-modules with integrable connections and
with Frobenius structures defined on the dagger unit polydisc
such that traces
of Frobenius on fibers at Techm\"uller points
define the hypergeometric function over the finite
field introduced by Gelfand and Graev.