Spatially and temporally adaptive algorithms can substantially improve the computational efficiency of many numerical schemes in computational mechanics and physics-based animation. Recently, a crucial need for temporal adaptivity in the Material Point Method (MPM) is emerging due to the potentially substantial variation of material stiffness and velocities in multimaterial scenes. In this work, we propose a novel temporally adaptive symplectic Euler scheme for MPM with regional time stepping (RTS), where different time steps are used in different regions. We design a time stepping scheduler operating at the granularity of small blocks to maintain a natural consistency with the hybrid particle/grid nature of MPM. Our method utilizes the Sparse Paged Grid (SPGrid) data structure and simultaneously offers high efficiency and notable ease of implementation with a practical multi-threaded particle-grid transfer strategy. We demonstrate the efficacy of our asynchronous MPM method on various examples including elastic objects, granular media, and fluids.