In cosmic ray air showers, the muon lateral separation from the center of the shower is a measure of the transverse momentum that the muon parent acquired in the cosmic ray interaction. IceCube has observed cosmic ray interactions that produce muons laterally separated by up to 400 m from the shower core, a factor of 6 larger distance than previous measurements. These muons originate in high p(T) (> 2 GeV/c) interactions from the incident cosmic ray, or high-energy secondary interactions. The separation distribution shows a transition to a power law at large values, indicating the presence of a hard p(T) component that can be described by perturbative quantum chromodynamics. However, the rates and the zenith angle distributions of these events are not well reproduced with the cosmic ray models tested here, even those that include charm interactions. This discrepancy may be explained by a larger fraction of kaons and charmed particles than is currently incorporated in the simulations. DOI: 10.1103/PhysRevD.87.012005