  In this paper, we consider the existence of static solutions to the nonlinear
Chern-Simons-Schr\"odinger system \begin{equation}\label{eqabstr}
\left\{\begin{array}{ll}
-ihD_0\Psi-h^2(D_1D_1+D_2D_2)\Psi+V\Psi=|\Psi|^{p-2}\Psi,\\
\partial_0A_1-\partial_1A_0=-\frac
12ih[\overline{\Psi}D_2\Psi-\Psi\overline{D_2\Psi}],\\
\partial_0A_2-\partial_2A_0=\frac
12ih[\overline{\Psi}D_1\Psi-\Psi\overline{D_1\Psi}],\\
\partial_1A_2-\partial_2A_1=-\frac12|\Psi|^2,\\ \end{array} \right.
\end{equation} where $p>2$ and non-radial potential $V(x)$ satisfies some
certain conditions. We show that for every positive integer $k$, there exists
$h_0>0$ such that for $0<h<h_0$, problem \eqref{eqabstr} has a nontrivial
static solution $(\Psi_h, A_0^h, A_1^h,A_2^h)$. Moreover, $\Psi_h$ is a
positive non-radial function with $k$ positive peaks, which approach to the
local maximum point of $V(x)$ as $h\to 0^+$.
