We present new H i spectral line imaging of the extremely metal-poor, star-forming dwarf irregular galaxy Leo P. Our H i images probe the global neutral gas properties and the local conditions of the interstellar medium (ISM). The H i morphology is slightly elongated along the optical major axis. We do not find obvious signatures of interaction or infalling gas at large spatial scales. The neutral gas disk shows obvious rotation, although the velocity dispersion is comparable to the rotation velocity. The rotation amplitude is estimated to be V c =15 ± 5 km s −1 . Within the H i radius probed by these observations, the mass ratio of gas to stars is roughly 2:1, while the ratio of the total mass to the baryonic mass is 15:1. We use this information to place Leo P on the baryonic Tully-Fisher relation, testing the baryonic content of cosmic structures in a sparsely populated portion of parameter space that has hitherto been occupied primarily by dwarf spheroidal galaxies. We detect the signature of two temperature components in the neutral ISM of Leo P; the cold and warm components have characteristic velocity widths of 4.2 ± 0.9 km s −1 and 10.1 ± 1.2 km s −1 , corresponding to kinetic temperature upper limits of ∼1100 K and ∼6200 K, respectively. The cold H i component is unresolved at a physical resolution of 200 pc. The highest H i surface densities are observed in close physical proximity to the single H ii region. A comparison of the neutral gas properties of Leo P with other extremely metal-deficient (XMD) galaxies reveals that Leo P has the lowest neutral gas mass of any known XMD, and that the dynamical mass of Leo P is more than two orders of magnitude smaller than any known XMD with comparable metallicity.