Recently there is renewed interest in quantum critical phase transitions (QCPT) at T = 0 K in metallic strongly correlated electron systems. From early experimental results, the QCPT in the Kondo-lattice compound YbRh$_2$Si$_2$ is not a case of the ordinary spin density wave (SDW) instability observed in Ce-based Kondo lattices, but a candidate for a novel locally critical case. Here, we observe that coexisting, static Fermi liquid (FL) and non-Fermi liquid (NFL) states are a key feature of the QCPT in YbRh$_2$Si$_2$. By means of nuclear magnetic resonance (NMR) spin–lattice relaxation time (T1) measurements on a single-crystalline sample, we find that the FL and NFL states are invariant, whereas their ratio in a crossover is field dependent near the QCPT. Such a pair of states has remained hidden in Ce compounds, owing presumably to the short lifetimes of the two states. We derive a scaling law for the occupation ratio of the two states, which could be widely applicable to Kondo-lattice systems.