Beam- and rod-like components are fundamental structural elements in countless application fields. Due to specific application requirements, these components frequently exhibit nonlinear behavior characterized by large deflections but small strains. The effective modeling of such structures is a non-trivial problem, as arbitrarily large 3D rotations are described by orthogonal transformations in the Euclidean space and constitute a non-commutative (Lie) group. An efficient strategy to handling finite rotations in this context is given by the so-called geometrically exact beam structural theories, initially introduced for the 2D case by Reissner and later refined and generalized by Simo in 3D. A variety of numerical methods have been developed within the computational mechanics community to model geometrically exact 3D beams based on Simo-Reissner theory. Ideally, large displacements analyses of shear-deformable 3D beams should be performed using discretization techniques involving a small number of unknowns having a clear physical meaning, and leading to objective, as well as locking- and singularity-free solutions for arbitrarily large rotations. In this work, we propose a novel virtual element (VE) formulation for geometrically exact 3D beams that is able to met all these desired features, extending the 2D Timoshenko beam VE developed by Wriggers. Our Simo-Reissner 3D beam VE introduces displacements and rotations at the element ends as the only nodal unknowns, even for higher-order ansatz interpolation functions. The accuracy and locking-free behavior of the proposed Simo-Reissner 3D beam VE are proved through a number of well-established benchmark numerical tests.