Firstly, let me give the details for $\ell_\infty(\aleph_0)$; $K$ stands for the closed unit ball of $\ell_\infty(\aleph_0)$. For every $n$ let $U_n={ x\in K: |x(n)| > 1/4 - \|x\| }$. The family $(U_n)$ is an open cover of $K$. Let $(F_m)$ be a partition of unity subordinate to $(U_n)$. For every $m$ let $n_m$ be the least integer $m$ such that $supp(F_m)$ is contained in $U_n$ and define $$F(x)=\sum_m n_m \cdot F_m(x)$$. Then using the arguments outlined above, one can show that $F(K\cap E)$ is unbounded for every infinite-dimensional subspace $E$ of $\ell_\infty(\aleph_0)$.
Secondly, let me remark that my argument for $\ell_\infty(\kappa)$ with $\kappa$ measurable is not correct; I apologize for that (I have a remark at the end). What I can show is that for every $\kappa$ (even measurable) there exists a continuous function $F:K_0\to\mathbb{R}$, where $K_0$ is the closed unit ball of $c_0(\kappa)$, such that $F(K_0\cap E)$ is unbounded for every infinite-dimensional subspace $E$ of $c_0(\kappa)$. The argument is a variation of the previous one. For every pair of rationals $0 < a < b < 1/4$ let $U_{a,b}$ be the set of all $x\in c_0(\kappa)$ such that for every $t\in\kappa$ either $|x(t)| < a$ or $|x(t)| > b$. Notice that $U_{a,b}$ is open in $K_0$ and for every $x\in K_0$ there exists such a pair $(a,b)$ such that $x\in U_{a,b}$. Now for every $n$ (including zero) and every pair $0 < a < b < 1/4$ let $U_{a,b,n}$ be the set of all $x\in U_{a,b}$ for which the cardinality of the set ${t: |x(t)| > b}$ is $n$. The family $(U_{a,b,n})$ is an open cover of $K_0$. Let $(F_i) (i\in I)$ be a partition of unity subordinate to a locally finite refinement of $(U_a,b,n)$. For every $i\in I$ set $L_i={n: there exist 0 < a < b < 1/4 s.t. supp(F_i) is contained in U_{a,b,n}}$ and let $n_i$ be the least element of $L_i$. Now define $F:K_0\to\mathbb{R}$ by $$F(x)=\sum_i n_i \cdot F_i(x)$$. It is continuous.
Now we check that $F(K_0\cap E)$ is unbounded for every infinite-dimensional subspace $E$ of $c_0(\kappa)$. So let $E$ be one. Since $c_0(\kappa)$ is hereditarily $c_0$, by James, we can find a normalized sequence $(e_n)$ in $E$ which a $2$-equivalent to the standard unit vector basis of $c_0$ (in particular, $(e_n)$ is weakly null). Fix some integer $M$. We may recursively select a sequence $(n_k)$ in $\mathbb{N}$ such that for all $k < m$ the sets ${t: |e_{n_k}(t)| > 1/4M}$ and ${t: |e_{n_m}(t)| > 1/4M}$ are disjoint. Consider that vector $e= \sum_{k=1}^M e_{n_k}$. Observe, first, that $1/2\leq \|e\| \leq 2$. Also notice that the set ${t: |e(t)|\geq 3/4}$ has cardinality at least $M$. Let us normalize $e$ and denote the normalized vector by $v$. The set ${ t: |v(t)| \geq 3/8 }$ has cardinality at least $M$. Let $i\in I$ be such that $F_i(v)>0$. Let $0 < a < b < 1/4$ and $n$ be arbitrary such that $supp(F_i)$ is contained in $U_{a,b,n}$. Notice that the set ${t: |v(t)| \geq 3/8}$ is contained in the set ${t: |v(t)|> b}$, and so, the cardinality of the set ${t: |v(t)| > b}$ is at least $M$. It follows that $n_i\geq M$ yielding that $F(v)\geq M$.
As far as I can tell, the question about $\ell_\infty(\kappa)$ with $\kappa\geq \aleph_1$ is open.
