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