  Frustrated magnets offer a plethora of exotic magnetic ground states,
including quantum spin liquids (QSLs), in which enhanced quantum fluctuations
prevent a long-range magnetic ordering of the strongly correlated spins down to
lowest temperature. Here we have investigated the trimer based mixed valence
hexagonal rhodate Ba$_4$NbRh$_3$O$_{12}$ using a combination of dc and ac
magnetization, electrical resistivity, specific heat, and muon spin
rotation/relaxation ($\mu$SR) measurements. Despite the substantial
antiferromagnetic exchange interactions, as evident from the Weiss temperature
($\theta_{\mathrm{W}}\sim -35$ to -45 K), among the Rh-local moments, neither
long-range magnetic ordering nor spin-freezing is observed down to at least 50
mK, in ac-susceptibility, specific heat and ZF-$\mu$SR measurements (down to
0.26 K). We ascribe the absence of any magnetic transition to enhanced quantum
fluctuations as a result of geometrical frustration arising out of the
edge-sharing equilateral Rh-triangular network in the structure. Our
longitudinal-field $\mu$SR result evidences persistent spin fluctuations down
to 0.26~K, thus stabilizing a dynamic QSL ground state in
Ba$_4$NbRh$_3$O$_{12}$. Furthermore, the magnetic specific heat
($C_{\mathrm{m}}$) data at low-$T$ reveal a significant $T$-linear contribution
plus a quadratic $T$-dependence. A $T$-linear behavior is evocative of gapless
spin excitations, while the $T^2$-term of $C_{\mathrm{m}}$ may indicate the
Dirac QSL phenomenology of the spinon excitations with a linear dispersion.
