The first amplitude analysis of the B±→π±K+K- decay is reported based on a data sample corresponding to an integrated luminosity of 3.0 fb-1 of pp collisions recorded in 2011 and 2012 with the LHCb detector. The data are found to be best described by a coherent sum of five resonant structures plus a nonresonant component and a contribution from ππ↔KK S-wave rescattering. The dominant contributions in the π± K and K+ K- systems are the nonresonant and the B±→ρ(1450)0π± amplitudes, respectively, with fit fractions around 30%. For the rescattering contribution, a sizable fit fraction is observed. This component has the largest CP asymmetry reported to date for a single amplitude of (-66±4±2)%, where the first uncertainty is statistical and the second systematic. No significant CP violation is observed in the other contributions.