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.
Firstly, let me recall that it is consistent that the the continuum is real-valued
measurable (R. M. Solovay). On the other hand, if CH holds, then there is heavy
(and quite advanced) machinery for ``killing" various Ramsey properties on
$\omega_1$ (largely due to S. Todorcevic).