  In 1970 Haskell Rosenthal proved that if $X$ is a Banach space, $\Gamma$ is
an infinite index set, and $T:\ell_\infty(\Gamma)\to X$ is a bounded linear
operator such that $\inf_{\gamma\in\Gamma}\|T(e_\gamma)\|>0$ then $T$ acts as
an isomorphism on $\ell_\infty(\Gamma')$, for some $\Gamma'\subset\Gamma$ of
the same cardinality as $\Gamma$. Our main result is a nonlinear strengthening
of this theorem. More precisely, under the assumption of GCH and the regularity
of $\Gamma$, we show that if ${F}:B_{\ell_\infty(\Gamma)}\to X$ is uniformly
differentiable and such that $\inf_{\gamma\in\Gamma}\|{F}(e_\gamma){-F(0)}\|>0$
then there exists $x\in B_{\ell_\infty(\Gamma)}$ such that $d{F}(x)[\cdot]$ is
a bounded linear operator which acts as an isomorphism on
$\ell_\infty(\Gamma')$, for some $\Gamma'\subset\Gamma$ of the same cardinality
as $\Gamma$.
