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From the answers to Mathoverflow Question "Continuity in Banach space for non-linear maps", it is possible to infer that the assumption of the Cauchy-Lipschitz theorem for the autonomous equation $$ \dot X=F(X), \tag{$\ast$} $$ could fail to be satisfied even if $F$ is $C^1$: in fact, using the mean-value inequality, the verification of the Cauchy-Lipschitz hypothesis seems to require local boundedness of the derivative, a property which could fail in infinite dimension even when this derivative is continuous.

My question: it is well-known that Peano's theorem (local existence with $F$ continuous, of course without uniqueness) fails in infinite dimensions. Is there a counterexample to existence or uniqueness for ($\ast$) when $F$ is $C^1$ (and of course $F'$ not locally bounded, otherwise CL theorem provides local existence and uniqueness)?