The velocity distribution function (VDF) of dark matter (DM) haloes in Λ cold dark matter (ΛCDM) dissipationless cosmological simulations, which must be non-separable in its radial and tangential components, is still poorly known. We present the first single-parameter, non-separable, anisotropic model for the VDF in ΛCDM haloes, built from an isotropic q-Gaussian (Tsallis) VDF of the isotropic set of dimensionless spherical velocity components (after subtraction of streaming motions), normalized by the respective velocity dispersions. We test our VDF on 90 cluster-mass haloes of a dissipationless cosmological simulation. Beyond the virial radius, r_vir, our model VDF adequately reproduces that measured in the simulated haloes, but no q-Gaussian model can adequately represent the VDF within r_vir, as the speed distribution function is then flatter-topped than any q-Gaussian can allow. Nevertheless, our VDF fits significantly better the simulations than the commonly used Maxwellian (Gaussian) distribution, at virtually all radii within 5 r_vir. Within 0.4 (1) r_vir, the non-Gaussianity index q is (roughly) linearly related to the slope of the density profile and also to the velocity anisotropy profile. We provide a parametrization of the modulation of q with radius for both the median fits and the fit of the stacked halo. At radii of a few per cent of r_vir, corresponding to the solar position in the Milky Way, our best-fitting VDF, although fitting better the simulations than the Gaussian one, overproduces significantly the fraction of high-velocity objects, indicating that one should not blindly use these q-Gaussian fits to make predictions on the direct detection rate of DM particles.