Carbonate minerals are of particular interest in paleoenvironmental research as they are an integral part of the carbon and water cycles, both of which are relevant to habitability. Given that these cycles are less constrained on Mars than they are on Earth, the identification of carbonates has been a point of emphasis for rover missions. Here, we present carbon (δ13C) and oxygen (δ18O) isotope data from four carbonates encountered by the Curiosity rover within the Gale crater. The carbon isotope values range from 72 ± 2‰ to 110 ± 3‰ Vienna Pee Dee Belemnite while the oxygen isotope values span from 59 ± 4‰ to 91 ± 4‰ Vienna Standard Mean Ocean Water (1 SE uncertainties). Notably, these values are isotopically heavy (13C- and 18O-enriched) relative to nearly every other Martian material. The extreme isotopic difference between the carbonates and other carbon- and oxygen-rich reservoirs on Mars cannot be reconciled by standard equilibrium carbonate-CO2 fractionation, thus requiring an alternative process during or prior to carbonate formation. This paper explores two processes capable of contributing to the isotopic enrichments: 1) evaporative-driven Rayleigh distillation and 2) kinetic isotope effects related to cryogenic precipitation. In isolation, each process cannot reproduce the observed carbonate isotope values; however, a combination of these processes represents the most likely source for the extreme isotopic enrichments.