  Properly discontinuous actions of a surface group by affine automorphisms of
$\mathbb R^d$ were shown to exist by Danciger-Gueritaud-Kassel. We show,
however, that if the linear part of an affine surface group action is in the
Hitchin component, then the action fails to be properly discontinuous. The key
case is that of linear part in $\mathsf{SO}(n,n-1)$, so that the affine action
is by isometries of a flat pseudo-Riemannian metric on $\mathbb R^d$ of
signature $(n,n-1)$. Here, the translational part determines a deformation of
the linear part into $\mathsf{PSO}(n,n)$-Hitchin representations and the
crucial step is to show that such representations are not Anosov in
$\mathsf{PSL}(2n,\mathbb R)$ with respect to the stabilizer of an $n$-plane. We
also prove a negative curvature analogue of the main result, that the action of
a surface group on the pseudo-Riemannian hyperbolic space of signature
$(n,n-1)$ by a $\mathsf{PSO}(n,n)$-Hitchin representation fails to be properly
discontinuous.
