  We study the scattering problem for the nonlinear Schr\"odinger equation
$i\partial_t u + \Delta u = |u|^p u$ on $\mathbb{R}^d$, $d\geq 1$, with a
mass-subcritical nonlinearity above the Strauss exponent. For this equation, it
is known that asymptotic completeness in $L^2$ with initial data in $\Sigma$
holds and the wave operator is well-defined on $\Sigma$. We show that there
exists $0<\beta<p$ such that the wave operator and the data-to-scattering-state
map do not admit extensions to maps $L^2\to L^2$ of class $C^{1+\beta}$ near
the origin. This constitutes a mild form of ill-posedness for the scattering
problem in the $L^2$ topology.
