  In this paper, we prove the existence of a family of non trivial compact
subdomains $\O$ in the manifold $\mathcal{M}=\R^N\times \R/2\pi\Z$ for which
the overdetermined Neumann boundary value problem
  \begin{align}\label{Neumann1}
  \left \{
  \begin{aligned}
  -\D w&=\mu |w|^{p-2}w && \qquad \text{in $ \Omega$,}
  \frac{\partial w}{\partial\eta} &=0 &&\qquad \text{on $\partial \Omega,$}
w&=c\ne 0 &&\qquad \text{on $\partial \Omega$,}
  \end{aligned}
  \right. \end{align} admits solutions for some $\mu > 0$ and
$2<p<2^{*}=\frac{2N}{N-2}$ if $N\geq 3$ or $p>2$ if $N=2$. The domains we
construct have nonconstant principal curvature, and therefore are not
isoparametric nor homogeneous. By this, we establish a non-linear analogue of a
recent result obtained by Fall, Weth and the first named author in
\cite{Fall-MinlendI-Weth4}, where the overdetermined Neumann eigenvalue problem
for the Laplacian was considered.
