Peano's theorem is false for any infinite-dimensional Fréchet space

It is proved that for any nonnormable Fréchet space $E$ a continuous map $f\colon E\to E$ and a closed infinite-dimensional subspace $L$ can be found such that the Cauchy problem $\dot x=f(x)$, $x(0)=u$ has no solution for any $u\in L$. Previous counterexamples to Peano's theorem cover Banach spaces and nonsemireflexive spaces.