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We present experimental and theoretical evidence for bond-dependent Kitaev-Γ-Heisenberg exchange in honeycomb Ni³⁺ magnet NaNi₂BiO_6-δ. Heat capacity and neutron diffraction reveal a low-temperature ordered magnetic state for T < 4.8 K with (1/3, 1/3, 0.15(1)) order. In-plane magnetic correlations match the 120° ordered state of the Kitaev-Γ-Heisenberg exchange. Electron spin resonance and density functional theory reveal a uniform 3+ valence in Ni intermediate between a S=3/2 and J=1/2 state. DFT shows that covalent bonding between oxygen and Bi produces an enhanced spin orbit coupling in Ni³⁺, which allows for strong bond-dependent exchange. The ordering wave vector, in-plane magnetic correlations, missing entropy, spin state, and superexchange pathways are all consistent with bond-dependent Kitaev-Γ-Heisenberg exchange interactions in NaNi₂BiO_6-δ.