An essential ingredient for the production of Majorana fermions for use in quantum computing is topological superconductivity^1,2. As bulk topological superconductors remain elusive, the most promising approaches exploit proximity-induced superconductivity³, making systems fragile and difficult to realize^4–7. Due to their intrinsic topology⁸, Weyl semimetals are also potential candidates^1,2, but have always been connected with bulk superconductivity, leaving the possibility of intrinsic superconductivity of their topological surface states, the Fermi arcs, practically without attention, even from the theory side. Here, by means of angle-resolved photoemission spectroscopy and ab initio calculations, we identify topological Fermi arcs on two opposing surfaces of the non-centrosymmetric Weyl material trigonal PtBi₂ (ref. ⁹). We show these states become superconducting at temperatures around 10 K. Remarkably, the corresponding coherence peaks appear as the strongest and sharpest excitations ever detected by photoemission from solids. Our findings indicate that superconductivity in PtBi₂ can occur exclusively at the surface, rendering it a possible platform to host Majorana modes in intrinsically topological superconductor–normal metal–superconductor Josephson junctions.