  In this paper, we consider the compressible Navier--Stokes system around the
constant equilibrium states and prove the unique existence of a global solution
for arbitrarily large initial data in the scaling critical Besov space provided
that the Mach number is sufficiently small and the incompressible part of the
initial velocity generates the global solution of the incompressible
Navier--Stokes equation. Moreover, we consider the low Mach number limit and
show that the compressible solution converges to the solution of the
incompressible Navier--Stokes equation in some space-time norms.
