  Kitaev's honeycomb spin-liquid model and its proposed realization in
materials such as $\alpha$-RuCl$_3$, Li$_2$IrO$_3$ and Na$_2$IrO$_3$ continue
to present open questions about how the dynamics of a spin-liquid are modified
in the presence of non-Kitaev interactions as well as the presence of
inhomogeneities. Here we use $^{23}$Na nuclear magnetic resonance to probe both
static and dynamical magnetic properties in single crystal Na$_2$IrO$_3$. We
find that the NMR shift follows the bulk susceptibility above 30 K but deviates
from it below; moreover below $T_N$ the spectra show a broad distribution of
internal magnetic fields. Both of these results provide evidence for
inequivalent magnetic sites at low temperature, suggesting inhomogeneities are
important for the magnetism. The spin-lattice relaxation rate is isotropic and
diverges at $T_N$, suggesting that the Kitaev cubic axes may control the
critical quantum spin fluctuations. In the ordered state, we observe gapless
excitations, which may arise from site substitution, emergent defects from
milder disorder, or possibly be associated with nearby quantum paramagnetic
states distinct from the Kitaev spin liquid.
