The similar to 86 ks ROSAT Position Sensitive Proportional Counter image of the Coma Cluster is deeper than any previous X-ray observation of a galaxy cluster. We search for X-ray emission from 35 individual galaxies in a magnitude-limited sample, all of which lie within 20' of the optical axis in at least one of the four Coma pointings. We detect seven galaxies in the 0.4-2.4 keV band at a significance level exceeding 3 sigma, and a further four at above 2 sigma. Although we can set only upper limits on the individual flux from each of the other galaxies, we are able to measure their mean flux by stacking the observations. The X-ray luminosities of the seven detections range from 6.2 x 10(40) to 1.5 x 10(42) ergs s(-1) (0.4-2.4 keV for H-0 = 50 km s(-1) Mpc(-1)). For galaxies with a blue absolute magnitude of about -21 we find a mean X-ray luminosity of 1.3 x 10(40) ergs s(-1). The ratio of X-ray to optical luminosity is substantially smaller for such objects than for the brightest galaxies in the cluster. The X-ray luminosities of the four brightest galaxies are ill-defined, however, because of ambiguity in distinguishing galaxy emission from cluster emission. Each object appears to be related to significant structure in the diffuse intracluster medium. We also investigate emission in the softer 0.2-0.4 keV band where detections are less significant because of the higher background, and we discuss the properties of a number of interesting individual sources. The X-ray luminosities of the Coma galaxies are similar to those of galaxies in the Virgo Cluster and in other regions with relatively low galaxy density. We conclude that large-scale environmental effects do not significantly enhance or suppress the average X-ray emission from galaxies, but that individual objects vary in luminosity substantially in a way which may depend on the detailed history of their local environment.