In this work we present a non-relativistic space--time formulation for geometrically exact beam structures [1]. The beam centreline is embedded into R⁴∼= R³ × R and interpreted as a two-dimensional world sheet, which enables a shell-like geometric description. Starting from Hamilton’s principle, we derive a variational formulation that consistently incorporates both elastic and inertial effects. The space--time world sheet is equipped with a covariant tangent basis induced by its parametric representation, providing the geometric ingredients required for an intrinsic (coordinate-consistent) formulation. Depending on the application, we employ either tensor-product shape functions (e.g.\ spline/NURBS spaces) or nonconforming discretizations (e.g.\ unstructured meshes), the latter enabling local adaptive refinement in space and/or time. To represent rotations consistently in space--time, the spatial rotation group SO(3) is embedded into SO(4). This admits spatial rotations while excluding SO(4) rotations that mix spatial and temporal directions, in accordance with the non-relativistic setting.The resulting restriction on the temporal component follows directly from the modelling assumption and can be interpreted as an implicit Dirichlet-type constraint on the time coordinate. We will demonstrate the applicability of the proposed method via a series of benchmark tests, using either analytical or numerical (overkill) solutions.