Counterexample to Peano's theorem in infinite-dimensional $F'$-spaces

Let $E$ be a nonnormable Fréchet space, and let $E'$ be the space of all continuous linear functionals on $E$ in the strong topology. A continuous mapping $f\colon E'\to E'$ such that for any $t_0\in\mathbb R,x_0\in E'$, the Cauchy problem $\dot x=f(x)$, $x(t_0)=x_0$ has no solutions is constructed.