In this study, the low-velocity impact response of a cylindrical sandwich shell with metal–fiber laminated (FML) face sheets and a compliant core is analytically investigated. The governing equations are derived using Hamilton’s principle within the framework of a higher-order sandwich shell theory and implemented in MATLAB for numerical computation. In the developed model, the face sheets are modeled based on the classical shell theory, while the displacement field of the core is described according to the three-dimensional elasticity theory without employing common simplifying assumptions. Moreover, nonlinear distributions of stresses and strains through the core thickness are incorporated as complementary relations, and both geometric nonlinearity and in-plane/out-of-plane shear deformations are included to achieve a more accurate description of the impact response. The time history of the contact force is validated against available results from previous studies, demonstrating good agreement. Furthermore, a parametric analysis is performed to assess the influence of key parameters such as the core thickness and elastic modulus, as well as the impactor mass, velocity, and radius on the impact behavior of the system. The results reveal that these parameters play a crucial role in determining the peak contact force and the overall deformation of the sandwich shells.