A We evaluate the dry merger activity in the Coma cluster, using a spectroscopically complete sample of 70 red-sequence (RS) galaxies, most of which (similar to 75%) are located within 0.2R(200) (similar to 0.5 Mpc) from the cluster center, with data from the Coma Treasury Survey obtained with the Hubble Space Telescope. The fraction of close galaxy pairs in the sample is the proxy employed for the estimation of the merger activity. We identify 5 pairs and 1 triplet, enclosing a total of 13 galaxies, based on limits on projected separation and line-of-sight velocity difference. Of these systems, none show signs of ongoing interaction, and therefore we do not find any true mergers in our sample. This negative result sets a 1 sigma upper limit of 1.5% per Gyr for the major dry merger rate, consistent with the low rates expected in present-day clusters. Detailed examination of the images of all the RS galaxies in the sample reveals only one with low surface brightness features identifiable as the remnant of a past merger or interaction, implying a post-merger fraction below 2%.