Evidence for direct<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:math>violation from Dalitz-plot analysis of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>B</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mi>K</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>∓</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>±</mml:mo></mml:msup></mml:math>

We report a Dalitz-plot analysis of the charmless hadronic decays of charged $B$ mesons to the final state ${K}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\pi}}^{\ensuremath{\mp}}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$. Using a sample of $(383.2\ifmmode\pm\else\textpm\fi{}4.2)\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $B\overline{B}$ pairs collected by the BABAR detector, we measure $CP$-averaged branching fractions and direct $CP$ asymmetries for intermediate resonant and nonresonant contributions. We find evidence for direct $CP$ violation in the decay ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{0}(770){K}^{\ifmmode\pm\else\textpm\fi{}}$, with a $CP$-violation parameter ${A}_{CP}=(+44\ifmmode\pm\else\textpm\fi{}10\ifmmode\pm\else\textpm\fi{}{4}_{\ensuremath{-}13}^{+5})%$.

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