**Does Peano's theorem apply to spaces with infinite dimension? Or is there a counterexample?**

Here, Peano's theorem is:

Let $E$ be a space with finite dimension. Consider a point $(t_0,x_0) \in \Re \times E$, constants $ a, b > $ 0 and a continuous function $$F: [t_0 - a, t_0 + a] \times B_b[x_0] \longrightarrow E$$ Then for every $ M> $ 0 satisfying $$\sup \{||F(t,x)||:(t,x) \in [t_0 - a, t_0 + a] \times B_b[x_0]\} < M$$ the Cauchy problem $$x'(t)=F(t,x(t));\ \ x(t_0)=x_0$$ admits at least one solution in the interval: $$\big[t_0 - \min(a,\frac{b}{M}),t_0 + \min(a,\frac{b}{M})\big] $$ An infinite-dimensional counterexample would be of great help. Thank you very much.