  We performed nuclear magnetic resonance studies on a kagome antiferromagnet
YCu$_3$(OH)$_6$Br$_2$[Br$_{1-x}$(OH)$_{x}$]. No significant NMR spectral
broadening is found in the Br center peak from 1 K down to 0.05 K, indicating
absence of static antiferromagnetic ordering. In contrast to signatures of
dominant 2D kagome antiferromagnetic fluctuations at temperature above 30 K,
both the Knight shift $K_{\rm{n}}$ and the spin-lattice relaxation rate
$1/T_{1}$ increase when the sample is cooled from 30 K to 8 K, which can be
attributed to the scattering of spin excitations by strong non-magnetic
impurities. Unusually, a hump is observed in $K_{\rm{n}}$ and $1/T_{2}$ close
to 2 K (far below the exchange energy), which indicates the existence of
excitations with a large density of states close to zero energy. These
phenomena are reproduced by a mean-field simulation of Heisenberg model with
bond-dependent exchange interactions, where the sign fluctuations in the spinon
kinetic terms caused by impurities result in localization of spinons and an
almost flat band close to the Fermi energy.
