In this paper, we consider the Chern–Simons–Schrödinger system \(\begin{aligned} \left\{ \begin{aligned}&-\varepsilon ^{2}\Delta u+V(x)u+(A_{0}+A_{1}^{2}+A_{2}^{2})u=|u|^{p-2}u,\,\,\,\,x\in \mathbb {R}^{2},\\&\partial _{1}A_{0}=A_{2}u^{2},\ \partial _{2}A_{0}=-A_{1}u^{2},\\&\partial _{1}A_{2}-\partial _{2}A_{1}=-\frac{1}{2}|u|^{2},\ \partial _{1}A_{1}+\partial _{2}A_{2}=0,\\ \end{aligned} \right. \end{aligned}\) where \(p>2,\) \(\varepsilon >0\) is a parameter and \(V:\mathbb {R}^{2}\rightarrow \mathbb {R}\) is a bounded continuous function. Under some mild assumptions on V(x), exploiting the finite-dimensional reduction method, we construct multi-peak solutions of the problem above. Also, we prove that all the concentrated solutions of the problem have the same form. Meanwhile, we present that the concentrated solutions are locally unique by various local Pohozaev identities, blow-up analysis and the maximum principle. Because of the nonlocal terms involved by \(A_{0},A_{1}\) and \(A_{2},\) we have to build a series of new and technical estimates which are very useful to study this problem.
