Cold spray is a rapidly developing coating technology for depositing materials in the solid state. In this work, the cold spray particle deposition process was simulated by modeling high-velocity impacts of spherical particles onto a flat substrate under various conditions. For the first time, we proposed the coupled Eulerian–Lagrangian (CEL) numerical approach as a means of solving the high-strain rate deformation problem. Using this approach, we observed a compressive stress region at the interface between the particles and the substrate induced by large plastic strains in the materials. Due to the high contact pressure (about 1 GPa) and the short contact time (about 40 ns), the high-strain rate ($106 s-1$) plastic deformation region was only a few micrometers deep and was localized mainly at the bottom of the particle and substrate surface. The ability of the CEL method to model the cold spray deposition process was assessed through a systematic parametric study including impact velocity, initial particle temperature, friction coefficient, and materials combination. The higher the impact velocity, the higher the initial kinetic energy, leading to more substantial plastic deformations and significant temperature increases in the substrate. The initial particle temperature has a greater influence on the equivalent plastic strain than on the temperature increase in the substrate. Friction has a limited effect on the temperature distribution and increase in the substrate, and the equivalent plastic strain increases only slightly as the friction coefficient rises. Four combinations of particle/substrate materials (Cu/Cu, Al/Al, Cu/Al, and Al/Cu) were considered in our study. Obviously, the particle's material had a greater influence on the deposition process and on the deformation than the substrate material. Concerning the particle's material, a higher-density material, such as Cu, has a higher initial kinetic energy, which has the advantage of increasing the contact area and contact time, resulting in better bonding between particles and substrate. Compared to other numerical methods (Lagrangian, arbitrary Lagrangian–Eulerian (ALE), and smooth particle hydrodynamics (SPH)), the CEL approach is globally more accurate and more robust in high-strain rate deformation regimes.