The CLARO-CMOS is a prototype ASIC that allows fast photon counting with 5 ns peaking time, a recovery time to baseline smaller than 25 ns, and a power consumption of less than 1 mW per channel. This chip is capable of single-photon counting with multi-anode photomultipliers and finds applications also in the read-out of silicon photomultipliers and microchannel plates. The prototype is realized in AMS 0.35 micron CMOS technology. In the LHCb RICH environment, assuming 10 years of operation at the nominal luminosity expected after the upgrade in Long Shutdown 2 (LS2), the ASIC must withstand a total fluence of about 610 super(12) 1 MeV n sub(eq)/cm super(2)neq/cm2 and a total ionizing dose of 400 krad. A systematic evaluation of the radiation effects on the CLARO-CMOS performance is therefore crucial to ensure long term stability of the electronics front-end. The results of multi-step irradiation tests with neutrons and X-rays up to the fluence of 10 super(14) cm super(-2) and a dose of 4 Mrad, respectively, are presented, including measurement of single event effects during irradiation and chip performance evaluation before and after each irradiation step.