  Let $\Gamma$ be an infinite set equipped with the discrete topology. We prove
that the space $\ell_{\infty}(\Gamma),$ of all complex-valued bounded functions
on $\Gamma$, satisfies the Mazur-Ulam property, that is, every surjective
isometry from the unit sphere of $\ell_{\infty}(\Gamma)$ onto the unit sphere
of an arbitrary complex Banach space $X$ admits a unique extension to a
surjective real linear isometry from $\ell_{\infty}(\Gamma)$ to $X$.
