  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 $x_c\approx 0.22$, 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.
