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 unique integer such that supp(F_m) 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).