  Consider the following nonlinear Schr\"odinger-Bopp-Podolsky system in
$\mathbb{R}^3$: $$ \begin{cases} -\varepsilon^2 \Delta u + (V + \phi) u = u
|u|^{p-1}; \\ a^2 \Delta^2 \phi - \Delta \phi = 4 \pi u^2, \end{cases} $$ where
$a, \varepsilon > 0$; $1 < p < 5$; $V \colon \mathbb{R}^3 \to ]0, \infty[$ and
we want to solve for $u, \phi \colon \mathbb{R}^3 \to \mathbb{R}$. By means of
Lyapunov-Schmidt reduction, we show that if $K \geq 2$, $z_0$ is a strict local
minimum of $V$, $V$ is adequately flat in a neighborhood of $z_0$ and
$\varepsilon$ is sufficiently small, then the system has a multipeak cluster
solution with $K$ peaks placed at the vertices of a regular convex $K$-gon
centered at $z_0$.
