(0.60 ≤ x ≤ 0.80)*

We report the results of our study of a recently synthesized honeycomb iridate Na_xIrO₃ (0.60 ≤ x ≤ 0.80). Single-crystal Na_xIrO₃ adopts a honeycomb lattice noticeably without distortions and stacking disorder inherently existent in its sister compound Na₂IrO₃. The oxidation state of the Ir ion is a mixed-valence state resulting from a majority Ir⁵⁺(5d⁴) ion and a minority Ir⁶⁺(5d³) ion. Na_xIrO₃ is a Mott insulator likely with a predominant pseudospin = 1 state. It exhibits an effective moment of 1.1μ_B/Ir and a Curie-Weiss temperature of −19 K but with no discernible long-range order above 1 K. The physical behavior below 1 K features two prominent anomalies at T_h = 0.9 K and T_l = 0.12 K in both the heat capacity and AC magnetic susceptibility. Intermediate between T_h and T_l lies a pronounced temperature linearity of the heat capacity with a large slope of 77 mJ/mole/K², a feature expected for highly correlated metals but not at all for insulators. These results along with a comparison drawn with the honeycomb lattices Na₂IrO₃ and (Na_0.2Li_0.8)₂IrO₃ point to an exotic ground state in proximity to a possible Kitaev spin liquid.