We present integral field spectroscopy of 10 early-type galaxies in the nearby, low-mass, Fornax cluster, from which we derive spatially resolved stellar kinematics. Based on the morphologies of their stellar velocity maps we classify 2/10 galaxies as slow rotators, with the remaining eight galaxies fast rotators. Supplementing our integral field observations with morphological and kinematic data from the literature, we analyse the 'kinematic' type of all 30 galaxies in the Fornax cluster brighter than MK = -21.5 mag (M* ~ 6 × 109 M⊙). Our sample's slow rotator fraction within one virial radius is 7-6+4 per cent. 13-6+8 per cent of the early-type galaxies are slow rotators, consistent with the observed fraction in other galaxy aggregates. The fraction of slow rotators in Fornax varies with cluster-centric radius, rising to 16-8+11 per cent of all kinematic types within the central 0.2 virial radii, from 0 per cent in the cluster outskirts. We find that, even in mass-matched samples of slow and fast rotators, slow rotators are found preferentially at higher projected environmental density than fast rotators. This demonstrates that dynamical friction alone cannot be responsible for the differing distributions of slow and fast rotators. For dynamical friction to play a significant role, slow rotators must reside in higher mass sub-haloes than fast rotators and/or form in the centres of groups before being accreted on to the cluster. © 2014 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society.