© 2014 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society. We present the measurements and modelling of the projected and redshift-space clustering of CMASS galaxies in the Sloan Digital Sky Survey-III Baryon Oscillation Spectroscopic Survey Data Release 11. For a volume-limited luminous red galaxy sample in the redshift range of 0.48 < z < 0.55, we perform halo occupation distribution modelling of the small- and intermediate-scale (0.1-60 h-1 Mpc) projected and redshift-space two-point correlation functions, with an accurate model built on high-resolution N-body simulations. To interpret the measured redshift-space distortions, the distribution of galaxy velocities must differ from that of the dark matter inside haloes of ~1013-1014 h-1 M⊙, i.e. the data require the existence of galaxy velocity bias. Most notably, central galaxies on average are not at rest with respect to the core of their host haloes, but rather move around it with a 1D velocity dispersion of 0.22-0.04+0.03 times that of the dark matter, implying a spatial offset from the centre at the level of ≲1 per cent of the halo virial radius. The luminous satellite galaxies move more slowly than the dark matter, with velocities 0.86-0.03+0.08 times those of the dark matter, which suggests that the velocity and spatial distributions of these satellites cannot both be unbiased. The constraints mainly arise from the Fingers-of-God effect at non-linear scales and the smoothing to the Kaiser effect in the translinear regime; the robustness of the results is demonstrated by a variety of tests. We discuss the implications of the existence of galaxy velocity bias for investigations of galaxy formation and cosmology.