In order to elucidate the effect of Ni ions in the Li layer on magnetism and Li diffusion of LiNiO₂, we have measured muon-spin rotation and muon-spin relaxation (μ⁺SR) spectra for the polycrystalline Li_1−xNi_1+xO₂ samples with x=0.02, 0.03, and 0.15. Weak transverse-field-μ⁺SR measurements demonstrated the existence of a bulk ferromagnetic transition at T_m=48(6) K for the x=0.03 sample and 161(7) K for x=0.15 while the x=0.02 sample exhibited an antiferromagnetic transition at 18(4) K. Zero-magnetic-field-(ZF) μ⁺SR measurements below T_m clarified the formation of static ferromagnetic (FM) order for the x=0.03 and 0.15 samples but only a highly disordered antiferromagnetic (AF) order for the x=0.02 sample. Therefore, the variation in the low-T magnetism with x is most unlikely due to the change in the concentration of an AF NiO-type domain or an FM Ni-rich cluster but likely due to a homogeneous change in the whole system. In the paramagnetic state, ZF- and longitudinal-field-μ⁺SR spectra exhibited a dynamic nuclear field relaxation. From the temperature dependence of the field fluctuation rate, a diffusion coefficient of Li⁺ ions (D_Li) at 300 K was estimated about 0.39(3)×10⁻¹¹ cm²/s for the x=0.02 sample and 0.12(7)×10⁻¹¹ cm²/s for x=0.15. On the other hand, the related compound, LiCrO2, did not show any diffusive behavior even at the highest temperature measured (=475 K). Considering the hindrance of diffusion by Ni in the Li⁺ diffusion plane and the fact that LiCrO₂ is electrochemically inactive, the estimated D_Li is thought to be very reasonable for the positive electrode material of Li-ion batteries. Furthermore, at low temperatures where the Li⁺ ions are static, the internal magnetic field was still found to be fluctuating, due to a dynamic local Jahn-Teller distortion of the Ni³⁺ ions in a low-spin state with S=1/2(t⁶_2ge¹_g).