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 such that
supp(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).