Several theorems regarding differential equations are true not only in finite dimension but also in infinite dimension Banach spaces. This is in particular the case for the **Cauchy–Lipschitz theorem**.

On the other end the Peano existence theorem is false for Banach space with infinite dimensions. See here for a counterexample.

Do you know other counterexamples in "classical" Banach spaces that are different from $c_0$ (the space of sequences of reals converging to $0$)? In particular, is there "an easy example" in the space $C([0,1],\mathbb{R})$ with $\sup$ norm?