  In this paper, we study a hyperbolic version of the Navier-Stokes equations,
obtained by using the approximation by relaxation of the Euler system, evolving
in a thin strip domain. The formal limit of these equations is a hyperbolic
Prandtl type equation, our goal is to prove the existence and uniqueness of a
global solution to these equations for analytic initial data in the tangential
variable, under a uniform smallness assumption. Then we justify the limit from
the anisotropic hyperbolic Navier-Stokes system to the hydrostatic hyperbolic
Navier-Stokes system with small analytic data.
