Using x-ray photon correlation spectroscopy, we studied the dynamics in the nematic phase of a nanorod suspension. The collective diffusion coefficient in the plane perpendicular to the director varies sharply with the wave vector. Combining the structure factor and the diffusion coefficient, we find that the hydrodynamic function of the phase decreases by more than a factor of 10 when going from length scales comparable to the interparticle distance toward larger values. Thus, the collective dynamics of the nematic phase experiences strong and scale-dependent slowing down, in contrast with isotropic suspensions of slender rods or of spherical particles.