We introduce an extension of the Higher Order Methods Modeling Environment (HOMME) dynamical core that solves the non-hydrostatic deep-atmosphere equations of motion. We call this extension NHD HOMME. Our extension satisfies analogs of the mimetic properties obeyed by the operational non-hydrostatic shallow-atmosphere (NHS) configuration of HOMME. Consequently, we demonstrate that our extension conserves energy under perfect temporal integration. We validate the software implementation of NHD HOMME using several idealized test cases. Small-earth steady-state experiments with a baroclinically unstable steady-state test case show that the horizontal discretizations in NHS HOMME and NHD HOMME are of comparable quality. Small-planet baroclinic wave tests match results from other deep-atmosphere dynamical cores in the literature. We find further evidence that baroclinic wave structure is sensitive to differences in the steady-state in different equation sets. In addition, we find that NHD HOMME produces small-planet Held-Suarez simulations that show an easterly equatorial zonal wind bias induced by discarding the Non-traditional Coriolis Terms (NCTs). These match results observed in other deep-atmosphere dynamical cores. Finally, simulations of the Matsuno-Gill idealized tropical heating forcing show that discarding the NCTs induces systematic errors in the linear response to large-scale heating on planets the size of earth. NHD HOMME will be provided as a switch-on option in the Energy Exascale Earth System Model.