The first amplitude analysis of the $B^{\pm }\to {\pi }^{\pm }{K}^{+}{K}^{-}$ decay is reported based on a data sample corresponding to an integrated luminosity of $3.0{\mathrm{fb}}^{-1}$ 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 $\pi \pi \leftrightarrow KK$ $S$-wave rescattering. The dominant contributions in the ${\pi }^{\pm }$ $K^{\mp }$ and $K^{+}$ $K^{-}$ systems are the nonresonant and the $B^{\pm }\to \rho (1450{)}^0{\pi }^{\pm }$ 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 $(-66\pm 4\pm 2)\%$, where the first uncertainty is statistical and the second systematic. No significant $CP$ violation is observed in the other contributions.