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The search for the experimental evidence of quantum spin liquid (QSL) states is critical but extremely challenging, as the quenched interaction randomness introduced by structural imperfection is usually inevitable in real materials. YCu$_3$(OH)$_{6.5}$Br$_{2.5}$ (YCOB) is a spin-1/2 kagome Heisenberg antiferromagnet (KHA) with strong coupling of $\langle J_1\rangle\sim$ 51 K but without conventional magnetic freezing down to 50 mK $\sim$ 0.001$\langle J_1\rangle$. Here, we report a Br nuclear magnetic resonance (NMR) study of the local spin susceptibility and dynamics on the single crystal of YCOB. The temperature dependence of NMR main-line shifts and broadening can be well understood within the frame of the KHA model with randomly distributed hexagons of alternate exchanges, compatible with the formation of a randomness-induced QSL state at low temperatures. The in-plane spin fluctuations as measured by the spin-lattice relaxation rates ($1/T_1$) exhibit a weak temperature dependence down to $T$ $\sim$ 0.03$\langle J_1\rangle$. Our results demonstrate that the majority of spins remain highly fluctuating at low temperatures despite the quenched disorder in YCOB.