The material point method (MPM) has been widely applied to various dynamic problems of solids and structures, including landslides and granular flows. Although explicit time integration schemes have been mainstream in MPM, an increasing number of studies employing implicit time integration schemes have been reported in recent years. However, most implicit MPM studies have focused on quasi-static problems, and only limited verification of accuracy and stability has been conducted for dynamic problems. In previous studies, the second-order accurate Newmark beta method has commonly been adopted as the time integration scheme. To suppress numerical instabilities arising from high-frequency components of the Newmark beta method and the small-mass issue inherent in MPM, techniques such as mass lumping and/or acceleration smoothing have often been introduced. While these techniques stably reproduce the dynamic behavior of solids, they are generally considered to introduce artificial energy dissipation in both high- and low-frequency components. In light of this background, the present study adopts the generalized-alpha method as the time integration scheme for dynamic implicit MPM in order to achieve both accuracy and stability. In the implicit MPM framework, extended B-spline basis functions are employed to prevent ill-conditioning of the stiffness matrix caused by small-mass issue. The generalized-alpha method is a second-order accurate algorithm that selectively attenuates high-frequency components. Through numerical examples, we demonstrate that the generalized-alpha method effectively improves numerical stability by damping the high-frequency components. We also show that stable computations can be achieved without introducing mass lumping or acceleration smoothing. Furthermore, in addition to the formulation using only displacement as the independent variable, a u-p mixed formulation, in which pressure is introduced as an additional independent variable, is also investigated. The applicability of this mixed formulation to dynamic analyses of compressible and nearly incompressible solids is evaluated.