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. Magnetic order 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 which again strongly indicates magnetic order in the low temperature regime. This is supported by our density functional theory calculations which show an appreciable Heisenberg exchange term in the spin Hamiltonian that favors 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 Neel and striped environments. A significant and undiminished dynamical relaxation rate ($sim 5$ MHz) as seen in $mu$SR deep into the ordered phase indicates enhanced quantum fluctuations in the ordered state.
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