Some properties of Ordinary Differential Equations - ODE are true in finite dimension spaces but not in Banach spaces of infinite dimension.

The first one I know is the Peano existence theorem. I give a counterexample here for infinite dimension.

The second one states that is the maximal solution of a differential equation is defined in an interval smaller than the one of definition of the map of the unique value problem, then the solution is "exploding". I give a counterexample here for infinite dimension.

Both are from the mathematician Jean Dieudonné.

Do you know other ODE properties valid in finite dimension spaces but not in Banach spaces of infinite dimension?