We report a $^{35}$Cl nuclear magnetic resonance (NMR) study of the diluted Kitaev material $alpha$-Ru$_{1-x}$Ir$_x$Cl$_3$ ($x=0.1$ and $0.2$) where non-magnetic Ir$^{3+}$ dopants substitute Ru$^{3+}$ ions. Upon dilution, the $^{35}$Cl spectra exhibit unusual large magnetic inhomogeneity, which sets in at temperatures below the Kitaev exchange energy scale. At the same time, the $^{35}$Cl spin-lattice relaxation rate $T_1^{-1}$ as a function of dilution and magnetic field unravels a critical doping of $x_capprox 0.22$, 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 $alpha$-Ru$_{1-x}$Ir$_x$Cl$_3$, arising from the interplay of dilution and exchange frustration in the quantum limit.
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