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