Peano's theorem in an infinite-dimensional Hilbert space is false even in a weakened formulation

We formulate a continuous function $F\colon R\times H\to H$, where $H$ is a separable Hilbert space such that the Cauchy problem
$$ x'(t)=F(t,x(t)),\quad x(t_0)=x_0 $$
has no solution in any neighborhood of the point $t_0$, no matter what $t_0\in R$ and $x_0\in H$ are considered.