Ady, I don't have an answer to the new version of your question but let me make some remarks which might be useful.
The new version is about non-linear real-valued continuous functions on $\ell_\infty(\Gamma)$ where $\Gamma$ has the cardinality of the continuum. This can be slightly generalized as follows:
Let $\kappa$ be an infinite cardinal and set $K$ to be the closed unit ball of $\ell_\infty(\kappa)$. Let $f:K\to\mathbb{R}$ be a continuous map. Does there exist an infinite-dimensional subspace $E$ of $\ell_\infty(\kappa)$ such that $f(K\cap E)$ is bounded?
If $\kappa=\aleph_0$, then a counterexample can be constructed.
On the other hand, if $\kappa$ is a measurable cardinal, then there exists a subspace $E$ of $\ell_\infty(\kappa)$ which is isomorphic to $c_0(\kappa)$ and such that $f(K\cap E)$ is bounded. The argument goes back to Ketonen. Let $FIN(\kappa)$ be the set of all non-empty finite subsets of $kappa$ and define a coloring $c:FIN(\kappa)\to\mathbb{N}$ as follows. Let $c(F)$ be $n$ if $n$ is the least integer $m$ such that
$ \max\{ |f(x)|: x\in span\{e_t: t\in F\} and x\in K \} \leq m $
where $e_t$ is the dirac function at $t$. Notice that $c$ is well-defined. There exist $n_0\in\mathbb{N}$ and a subset $A$ of $\kappa$ with $|A|=\kappa$ and such that $c$ is constant on $FIN(A)$ and equal to $n_0$. If we set $E$ to be the closed linear span of $\{e_t: t\in A\}$, then $E$ is isomorphic to $c_0(\kappa)$ and $F(K\cap E)$ is in the interval $[-n_0, n_0]$.
Concerning the continuum: it might be that there are set-theoretic issues. Let me recall that it is consistent that the the continuum is real-valued measurable (R. M. Solovay).