A new pressure-based lattice-Boltzmann method (HRR-p) is proposed for the simulation of flows for Mach numbers ranging from 0 to 1.5. Compatible with nearest-neighbor lattices (e.g., D3Q19), the model consists of a predictor step comparable to classical athermal lattice-Boltzmann methods, appended with a fully local and explicit correction step for the pressure. Energy conservation—for which the Hermitian quadrature is not accurate enough on such a lattice—is solved via a classical finite volume MUSCL-Hancock scheme based on the entropy equation. The Euler part of the model is then validated for the transport of three canonical modes (vortex, entropy, and acoustic propagation), while its diffusive/viscous properties are assessed via thermal Couette flow simulations. All results match the analytical solutions with very limited dissipation. Last, the robustness of the method is tested in a one-dimensional shock tube and a two-dimensional shock–vortex interaction.