In this work, a newly developed Smoothed Particle Hydrodynamics (SPH) algorithm for nonlinear elasticity is combined with an incompressible SPH fluid solver to investigate the dynamics of a floating plate under impacts of regular water waves with a high steepness. Two scenarios of the plate's rigidity are investigated. The simulation results show that deformations of the stiffer plate mainly occur in a simple bending mode with small amplitudes, and the plate is almost submerged by a strong fluid flow over its surface. In the other scenario, the plate deforms more complexly with much higher deformation amplitudes but experiences a much weaker overwash. The more flexible plate is less resistant to wave motions and converts more wave energy into elastic deformations, and therefore, the overwash is less severe. A strong overflow exerts a pressure force onto the plate that alters the plate's dynamics and adds a viscous (damping) effect on the plate's elastic vibrations, especially in high-frequency modes. A rigorous examination of the numerical convergence and validation using the linear thin plate theory is also carried out. The new SPH algorithm for nonlinear elasticity shows its stability and reliability in evaluating finite and large elastic deformations. Therefore, it is promising for simulating elastic structures in fluid-structure interaction problems.