Using kinematic maps from the Sloan Digital Sky Survey (SDSS) Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) survey, we reveal that the majority of low-mass quenched galaxies exhibit coherent rotation in their stellar kinematics. Our sample includes all 39 quenched low-mass galaxies observed in the first year of MaNGA. The galaxies are selected with Mr > -19.1, stellar masses 109 M⊙ < M* < 5 × 109 M⊙, EWHα < 2 Å, and all have red colours (u - r) > 1.9. They lie on the size-magnitude and σ-luminosity relations for previously studied dwarf galaxies. Just six (15 ± 5.7 per cent) are found to have rotation speeds ve, rot < 15 km s-1 at ˜1 Re, and may be dominated by pressure support at all radii. Two galaxies in our sample have kinematically distinct cores in their stellar component, likely the result of accretion. Six contain ionized gas despite not hosting ongoing star formation, and this gas is typically kinematically misaligned from their stellar component. This is the first large-scale Integral Field Unit (IFU) study of low-mass galaxies selected without bias against low-density environments. Nevertheless, we find the majority of these galaxies are within ˜1.5 Mpc of a bright neighbour (MK < -23; or M* > 5 × 1010 M⊙), supporting the hypothesis that galaxy-galaxy or galaxy-group interactions quench star formation in low-mass galaxies. The local bright galaxy density for our sample is ρproj = 8.2 ± 2.0 Mpc-2, compared to ρproj = 2.1 ± 0.4 Mpc-2 for a star-forming comparison sample, confirming that the quenched low-mass galaxies are preferentially found in higher density environments.