In a strongly interacting Rydberg atom array, the dynamics are often constrained to the decoupled and fragmented Hilbert subspaces, representing an intriguing paradigm for nonergodicity. Here we investigate the Hilbert-space fragmentation in the degenerate subspaces arising from the interplay between tunable detuning and arbitrary long-range interactions in a one-dimensional Rydberg-atom array with lattice-translation symmetry. We analytically establish the conditions for the emergence of subspaces and their subsidiary fragmentations in the thermodynamic limit, enumerate the emergent kinetic constraints at the discrete dynamical points where dynamics are possible at all, and demonstrate the conditions for strong fragmentation and the rich scaling behaviors of the fragments. Our results provide systematic and analytical understandings of Hilbert-space fragmentation in Rydberg arrays, and shed light on engineering nonergodic many-body dynamics beyond the PXP model.