Ady, I think there is a counterexample to your question. To describe it, let $(V_n)$ be a basis of $[0,1]$ consisting of non-empy open sets; $K$ stands for the closed unit ball of $C[0,1]$. For every $n$ let $C_n$ be the closure of $V_n$ and define
$U_n={g \in K: \min{|g(t)|:t \in C_n} > \|g\| - 1/4}$
where $\|g\|=\sup{|g(t)|:t \in [0,1]}$.
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 $n$ such that $\sup(F_m)={g \in K: F_m(g)>0}$ is contained in $U_n$.
Now define $F:K\to \mathbb{R}$ by
$F(g)= \sum_{m=1}^{\infty} n_m\cdot F_m(g)$
Notice that F is well-defined and continuous.
Finally notice that $F(K\cap E)$ is unbounded for every infinite-dimensional subspace E of $C[0,1]$. This follows from the following fact: for every integer i and every infinte-dimensional subspace $E$ of $C[0,1]$ there is a norm-one vector $e \in E$ such $e$ is NOT in $U_n$ for every $n < i$ (and therefore, if $m$ is such that $F_m(e)>0$, then necessarily $n_m$ is greater or equal to $i$ which gives that $F(e)$ is also greater or equal to $i$).
