In this study, we develop a mixed formulation in which the displacement and pressure of the solid phase are treated as independent variables within the framework of finite deformation elastoplasticity. This formulation is incorporated into the implicit MPM, aiming to stabilize the pressure field in the solid phase and validate its performance through analyses based on the modified Cam-Clay model. In elastoplastic models commonly used for geomaterials, pressure oscillations often arise in both the solid phase and the pore water due to the nearly incompressible nature of the material. To address this issue specifically in the solid phase, we develop a displacement-pressure mixed formulation within the framework of finite deformation elastoplasticity. To satisfy the stability condition in the mixed formulation, known as the LBB condition, we employ the variational multiscale (VMS) method, which allows the use of equal-order basis functions for both fields. As a constitutive model, we adopt the modified Cam-Clay model based on a two-invariant stored energy function. Furthermore, the proposed stabilized mixed formulation is implemented into the implicit MPM based on extended B-splines, which ensures stable computations with implicit time integration and effectively mitigates cell-crossing instability. Through large deformation analysis of clay, we demonstrate the effectiveness of the proposed method in suppressing pressure oscillations.