© 2020 American Physical Society. US. Kitaev's honeycomb spin-liquid model and its proposed realization in materials such as α-RuCl3, Li2IrO3, and Na2IrO3 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 Na23 nuclear magnetic resonance to probe both static and dynamical magnetic properties in single-crystal Na2IrO3. We find that the NMR shift follows the bulk susceptibility above 30 K but deviates from it below; moreover below TN 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 TN, 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.