The very high theoretical specific energy of the lithium–air (Li–O2) battery (3500 Wh kg−1) compared with other batteries makes it potentially attractive, especially for the electrification of flight. While progress has been made in realizing the Li–air battery, several challenges remain. One such challenge is achieving a high capacity to store charge at the positive electrode at practical current densities, without which Li–air batteries will not outperform lithium‐ion. The capacity is limited by the mass transport of O2 throughout the porous carbon positive electrode. Here it is shown that by replacing the binder in the electrode by a polymer with the intrinsic ability to transport O2, it is possible to reach capacities as high as 31 mAh cm−2 at 1 mA cm−2 in a 300 µm thick electrode. This corresponds to a positive electrode energy density of 2650 Wh L−1 and specific energy of 1716 Wh kg−1, exceeding significantly Li‐ion batteries and previously reported Li–O2 cells. Due to the enhanced oxygen diffusion imparted by the gas diffusion polymer, Li2O2 (the product of O2 reduction on discharge) fills a greater volume fraction of the electrode and is more homogeneously distributed.