III-V semiconductor nanowire arrays show promise as a platform for next-generation solar cells. However, the theoretical efficiency limit for converting the energy of sunlight into electrical energy in such solar cells is unknown. Here, we calculate through electromagnetic modeling the Shockley-Queisser efficiency limit for an InP nanowire array solar cell. In this analysis, we calculate first from the absorption of sunlight the short-circuit current. Next, we calculate the voltage-dependent emission characteristics of the nanowire array. From these processes, we identify how much current we can extract at a given voltage. Finally, after constructing this current-voltage (IV) curve of the nanowire solar cell, we identify from the maximum power output the maximum efficiency. We compare this efficiency of the nanowire array with the 31.0% efficiency limit of the conventional InP bulk solar cell with an inactive substrate underneath. We consider a nanowire array of 400 nm in period, which shows a high short-circuit current. We optimize both the nanowire length and diameter in our analysis. For example, nanowires of 4 μm in length and 170 nm in diameter produce 96% of the short-circuit current obtainable in the perfectly absorbing InP bulk cell. However, the nanowire solar cell emits fewer photons than the bulk cell at thermal equilibrium, especially into the substrate. This weaker emission allows for a higher open circuit-voltage for the nanowire cell. As an end result, nanowires longer than 4 μm can actually show, despite producing a lower short-circuit current, a higher efficiency limit, of up to 32.5%, than the bulk cell.