The material point method (MPM) aims to avoid mesh distortion problems occurring in the commonly used finite element method (FEM). To achieve this, the continuum is discretized as a set of material points, while the solution of the weak form of the underlying partial differential equations is conducted on an Eulerian background mesh. As this process works similar to FEM, the MPM also displays volumetric locking for near‐incompressible materials. However, this also means that mitigation techniques developed for FEM can be adapted to the MPM. The main difficulty for this process lies in the higher‐continuity basis functions, which are required in MPM to achieve reasonable results. This study first reviews the applicability of different locking mitigation techniques in MPM. Then, a framework for a locking‐free implicit MPM is presented and validated using common tests investigating the locking behavior and stress oscillations. Finally, the presented methods are applied to a viscoelastic material model representing unvulcanized rubber to show the potential of MPM for tire molding simulations.