  We consider a periodic higher-order nonlinear Schr\"odinger equation with the
nonlinearity $u^k \partial_x u$, where $k$ is a natural number. We prove the
norm inflation in a subspace of the Sobolev space $H^s(\mathbb{T})$ for any $s
\in \mathbb{R}$. In particular, the Cauchy problem is ill-posed in
$H^s(\mathbb{T})$ for any $s \in \mathbb{R}$.
