  Recently, there have been contrary claims of Kitaev spin-liquid behaviour and
ordered behavior in the honeycomb compound Ag$_3$LiIr$_2$O$_6$ based on various
experimental signatures. Our investigations on this system reveal a
low-temperature ordered state with persistent dynamics down to the lowest
temperatures. This is confirmed by clear oscillations in the muon spin
relaxation ($\mu$SR) time spectrum below 9 K till 52 mK. Coincidentally in
$^7$Li nuclear magnetic resonance, a wipe-out of the signal is observed below
$\sim$10 K strongly indicating magnetic order. This is supported by our density
functional theory calculations which show significant Heisenberg terms in the
spin Hamiltonian that favor magnetic ordering. The $^7$Li shift and
spin-lattice relaxation rate also show anomalies at $\sim$50 K. They are likely
related to the onset of dynamic magnetic correlations, but their origin is not
completely clear. Detailed analysis of our $\mu$SR data is consistent with a
co-existence of incommensurate N\'eel and striped environments. A significant
dynamical relaxation rate in the ordered phase as seen in $\mu$SR indicates
enhanced quantum fluctuations in the ordered state.
