The simulation of viscoplastic fluids, especially in scenarios involving interactions between different materials (such as granular flows coupled with fluid flows), presents significant challenges due to large deformations and the presence of free surfaces. To address this, we propose a stabilized mixed formulation for the implicit Material Point Method (MPM), tailored for viscoplastic flows. The formulation is inspired by classical non- linear solid mechanics approaches, making it particularly suitable for problems where fluid and solid behaviors coexist, through the careful selection of an appropriate constitutive law. However, when simulating fluid flows such as water, the problem enters an incompress- ible regime. In this case, standard Galerkin displacement-based formulations become unstable due to the violation of the inf-sup condition. To overcome this, we introduce a mixed displacement-pressure formulation stabilized using a Variational Multiscale (VMS) approach, which ensures both stability and accuracy in the pressure field. The proposed method has been validated through a series of numerical benchmarks de- signed to test its robustness, including simulations of saturated granular flows and water- laden landslides. These tests highlight the capability of the stabilized mixed MPM for- mulation to accurately capture complex flow dynamics and demonstrate its potential as a robust tool for multi-material simulations involving large deformations and free-surface phenomena.