We report the signatures of dynamic spin fluctuations in the layered honeycomb Li$_3$Cu$_2$SbO$_6$ compound, with a 3$d$ S = 1/2 $d^9$ Cu$^{2+}$ configuration, through muon spin rotation and relaxation ($mu$SR) and neutron scattering studies. Our zero-field (ZF) and longitudinal-field (LF)-$mu$SR results demonstrate the slowing down of the Cu$^{2+}$ spin fluctuations below 4.0 K. The saturation of the ZF relaxation rate at low temperature, together with its weak dependence on the longitudinal field between 0 and 3.2 kG, indicates the presence of dynamic spin fluctuations persisting even at 80 mK without static order. Neutron scattering study reveals the gaped magnetic excitations with three modes at 7.7, 13.5 and 33 meV. Our DFT calculations reveal that the next nearest neighbors (NNN) AFM exchange ($J_{AFM}$ = 31 meV) is stronger than the NN FM exchange ($J_{FM}$ = -21 meV) indicating the importance of the orbital degrees of freedom. Our results suggest that the physics of Li$_3$Cu$_2$SbO$_6$ can be explained by an alternating AFM chain rather than the honeycomb lattice.
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