We report a Cl35 nuclear magnetic resonance (NMR) study of the diluted Kitaev material α-Ru1-xIrxCl3 (x=0.1 and 0.2) where nonmagnetic Ir3+ dopants substitute Ru3+ ions. Upon dilution, the Cl35 spectra exhibit unusual large magnetic inhomogeneity, which sets in at temperatures below the Kitaev exchange energy scale. At the same time, the Cl35 spin-lattice relaxation rate T1-1 as a function of dilution and magnetic field unravels a critical doping of xc≈0.22, towards which both the field-induced spin gap and the zero-field magnetic ordering are simultaneously suppressed, while novel gapless low-energy spin excitations dominate the relaxation process. These NMR findings point to the stabilization of a random singlet phase in α-Ru1-xIrxCl3, arising from the interplay of dilution and exchange frustration in the quantum limit.