Dedicated H I surveys have recently led to a growing category of low-mass galaxies found in the Local Volume. We present synthesis imaging of one such galaxy, Pisces A, a low-mass dwarf originally confirmed via optical imaging and spectroscopy of neutral hydrogen (H I) sources in the Galactic Arecibo L-band Feed Array H I survey. Using H I observations taken with the Karl G. Jansky Very Large Array, we characterize the kinematic structure of the gas and connect it to the galaxy's environment and evolutionary history. While the galaxy shows overall ordered rotation, a number of kinematic features indicate a disturbed gas morphology. These features are suggestive of a tumultuous recent history and represent ∼3.5% of the total baryonic mass. We find a total baryon fraction fbary=0.13 if we include these features. We also quantify the cosmic environment of Pisces A, finding an apparent alignment of the disturbed gas with nearby, large-scale filamentary structure at the edge of the Local Void. We consider several scenarios for the origin of the disturbed gas, including gas stripping via ram pressure or galaxy-galaxy interactions, as well as accretion and ram pressure compression. Though we cannot rule out a past interaction with a companion, our observations best support the suggestion that the neutral gas morphology and recent star formation in Pisces A are a direct result of its interactions with the intergalactic medium.