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### Description

We present the first theoretical calculation of nonfactorizable charm-quark loop contributions to the $B_s\to \gamma\, l^+l^-$ amplitude. This contribution involves the $B$-meson three-particle Bethe-Salpeter amplitude, $\langle 0|\bar s(y)G_{\mu\nu}(x)b(0)|\bar B_s(p)\rangle$, for which we take into account constraints from analyticity and continuity. We calculate the relevant form factors, $H_{A,V}^{\rm NF}(k'^2,k^2)$, and provide convenient parametrizations of our results, applicable in the region below hadron resonances, $k'^2 < M_{J/\psi}^2$ and $k^2 < M_{\phi}^2$. We report that factorizable and nonfactorizable charm contributions to the $B_s\to\gamma\, l^+l^-$ amplitude have opposite signs. To compare the charm and the top contributions, the nonfactorizable charming loop contribution is expressed as a non-universal (i.e., dependent on the reaction) $q^2$-dependent correction $\Delta^{\rm NF}C_7(q^2)$ to the Wilson coefficient $C_7$. For the $B_s\to\gamma\, l^+l^-$ amplitude the correction is found to be positive, $\Delta^{\rm NF} C_7(q^2)/C_7 > 0$. Our numerical results for the form factors $H^{\rm NF}_i(k'^2,k^2)$ depend sizeably on the precise value of the parameter $\lambda_{B_s}$, and for a fixed value of $\lambda_{B_s}$, $H^{\rm NF}_i(k'^2,k^2)$ may be calculated with about 10% accuracy.