  We study the Sobolev critical Schr\"odinger-Bopp-Podolsky system
\begin{gather*}
  -\Delta u+\phi u=\lambda u+\mu|u|^{p-2}u+|u|^4u\quad \text{in }\mathbb{R}^3,
  -\Delta\phi+\Delta^2\phi=4\pi u^2\quad \text{in } \mathbb{R}^3, \end{gather*}
under the mass constraint \[
  \int_{\mathbb{R}^3}u^2\,dx=c \] for some prescribed $c>0$, where $2<p<8/3$,
$\mu>0$ is a parameter, and $\lambda\in\mathbb{R}$ is a Lagrange multiplier. By
developing a constraint minimizing approach, we show that the above system
admits a local minimizer. Furthermore, we establish the existence of normalized
ground state solutions.
