The results of a search for hydrogen-like atoms consisting of $pi^{mp}K^{pm}$ mesons are presented. Evidence for $pi K$ atom production by 24 GeV/c protons from CERN PS interacting with a nickel target has been seen in terms of characteristic $pi K$ pairs from their breakup in the same target ($178 pm 49$) and from Coulomb final state interaction ($653 pm 42$). Using these results the analysis yields a first value for the $pi K$ atom lifetime of $tau=(2.5_{-1.8}^{+3.0})$ fs and a first model-independent measurement of the S-wave isospin-odd $pi K$ scattering length $left|a_0^-right|=frac{1}{3}left|a_{1/2}-a_{3/2}right|= left(0.11_{-0.04}^{+0.09} right)M_{pi}^{-1}$ ($a_I$ for isospin $I$).