The dynamic simulation of slender structures such as beams plays a key role in many engineering applications, including aerospace structures, flexible robotics, and biomechanical systems. Conventional approaches typically rely on spatial discretization using the finite element method (FEM) combined with time-stepping schemes such as the Newmark algorithm. In contrast, space-time finite element formulations apply FEM simultaneously in both space and time, resulting in a single system that represents the complete dynamic evolution of the problem. Building on recent developments in space-time finite element methods for solid mechanics, we present a fully nonlinear space-time formulation for three-dimensional geometrically exact Simo--Reissner beams. A mixed formulation is employed in which displacements and velocities are discretized independently, allowing for a straightforward imposition of initial and boundary conditions while providing flexibility in the choice of interpolation spaces. Numerical stability and robustness are enhanced through the use of a residual-based stabilization technique. The beam is modeled as a Cosserat continuum, requiring both position and orientation fields on the space-time surface to be treated as unknowns. Finite rotations are consistently handled using a nodal rotation (pseudo-)vector description of the orientation field, combined with established rotational interpolation schemes that ensure objectivity. In the final discretized space-time problem, each finite element node possesses twelve degrees of freedom: translational displacements, rotational vectors, translational velocities, and material angular velocities. Selected qualitative and quantitative examples demonstrate the accuracy and objectivity of the proposed method and highlight its potential for applications in nonlinear structural dynamics, particularly in the presence of changing material domains.