  Quantum spin liquids are elusive but paradigmatic examples of strongly
correlated quantum states that are characterized by long-range quantum
entanglement. Recently, the direct signatures of a gapped topological
$\mathbb{Z}_2$ spin liquid have been observed in a system of Rydberg atoms
arrayed on the ruby lattice. Here, we illustrate the concrete realization of a
fundamentally different class of spin liquids in a honeycomb array of Rydberg
atoms. Exploring the quantum phase diagram of this system using both
density-matrix renormalization group and exact diagonalization simulations,
several density-wave-ordered phases are characterized and their origins
explained. More interestingly, in the regime where third-nearest-neighbor atoms
lie within the Rydberg blockade radius, we find a novel ground state -- with an
emergent $\mathrm{U}(1)\times \mathrm{U}(1)$ local symmetry -- formed from
superpositions of classical {\it trimer} configurations on the dual triangular
lattice. The fidelity of this trimer spin liquid state can be enhanced via
dynamical preparation, which we explain by a Rydberg-blockade-based projection
mechanism associated with the smooth turnoff of the laser drive. Finally, we
discuss the robustness of the trimer spin liquid phase under realistic
experimental parameters and demonstrate that our proposal can be readily
implemented in current Rydberg atom quantum simulators.
