A nice survey is the paper of Lobanov, and Smolyanov:
Sergey Grigorievich, Lobanov and Oleg Georgievich Smolyanov. ", Ordinary differential equations in locally convex spaces.Ordinary differential equations in locally convex spaces." Russian Mathematical Surveys 49.3 (1994): 97-17597–175.
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.