We prove the strong unique continuation property for many-body Pauli operators with external potentials, interaction potentials and magnetic fields in $L^p_{\rm loc}(\mathbb{R}^d)$, and with magnetic potentials in $L^q_{\rm loc}(\mathbb{R}^d)$, where $p > \max(2d/3,2)$ and $q > 2d$. For this purpose, we prove a singular Carleman estimate involving fractional Laplacian operators. Consequently, we obtain the Hohenberg-Kohn theorem for the Maxwell-Schr\"odinger model.