ODE properties true in finite dimension but not in Banach spaces of infinite dimension

A nice survey is the paper of Lobanov and Smolyanov:

Sergey Grigorievich Lobanov and Oleg Georgievich Smolyanov, Ordinary differential equations in locally convex spaces. Russian Mathematical Surveys 49.3 (1994): 97–175.

The paper lists several counterexamples in the infinite dimensional setting for classical properties and theorems of ODEs in finite-dimensional spaces, such as Peano's theorem, Kneser's theorem, continuous dependence on initial data, continuation of solutions and Picard's theorem.

You can find some relevant information in Godunov, A. N. The Peano theorem in Banach spaces. Functional Anal. Appl. 9 (1975), no. 1, 53–55 and Pasika, E. E. An example of a first-order differential equation in a Hilbert space without continuous dependence of the solution on the initial condition. (Russian) Ukrain. Mat. Zh. 35 (1983), no. 6, 786–788. Also you can check papers citing Godunov's paper in MathSciNet or Google Scholar.