The Mazur-Ulam property for the space of complex null sequences

Given an infinite set $\mathrm{\Gamma }$, we prove that the space of complex null sequences $c_0(\mathrm{\Gamma })$ satisfies the Mazur-Ulam property, that is, for each Banach space $X$, every surjective isometry from the unit sphere of $c_0(\mathrm{\Gamma })$ onto the unit sphere of $X$ admits a (unique) extension to a surjective real linear isometry from $c_0(\mathrm{\Gamma })$ to $X$. We also prove that the same conclusion holds for the finite dimensional space ${\mathrm{\ell }}_{\mathrm{\infty }}^m$.