As a demonstration of the capabilities of the new Oxford SWIFT integral field spectrograph, we present first observations for a set of 14 early-type galaxies in the core of the Coma cluster. Our data consist of I- and z-band spatially resolved spectroscopy obtained with the Oxford SWIFT spectrograph, combined with r-band photometry from the Sloan Digital Sky Survey archive for 14 early-type galaxies. We derive spatially resolved kinematics for all objects from observations of the calcium triplet absorption features at ∼8500 Å. Using this kinematic information we classify galaxies as either fast rotators or slow rotators. We compare the fraction of fast and slow rotators in our sample, representing the densest environment in the nearby Universe, to results from the ATLAS3D survey, finding that the slow rotator fraction is ∼50 per cent larger in the core of the Coma cluster than in the volume-limited ATLAS3D sample, a 1.2σ increase given our selection criteria. Comparing our sample to the Virgo cluster core only (which is 24 times less dense than the Coma core) we find no evidence of an increase in the slow rotator fraction. Combining measurements of the effective velocity dispersion σe with the photometric data we determine the Fundamental Plane for our sample of galaxies. We find that the use of the average velocity dispersion within 1 effective radius, σe, reduces the residuals by 13 per cent with respect to comparable studies using central velocity dispersions, consistent with other recent integral field Fundamental Plane determinations.