The variation in mean-square nuclear charge radius of xenon isotopes was measured utilizing a new method based on extreme ultraviolet spectroscopy of highly charged Na-like ions. The isotope shift of the Na-like D1 (3s $^{2}$S$_{1/2}$ - 3p $^2$P$_{1/2}$) transition between the $^{124}$Xe and $^{136}$Xe isotopes was experimentally determined using the electron beam ion trap facility at the National Institute of Standards and Technology. The mass shift and the field shift coefficients were calculated with enhanced precision by relativistic many-body perturbation theory and multi-configuration Dirac-Hartree-Fock method. The mean-square nuclear charge radius difference was found to be $delta<r^2>^{136, 124}$ = 0.269(0.042) fm$^2$. Our result has smaller uncertainty than previous experimental results and agrees with the recommended value by Angeli and Marinova [I. Angeli and K. P. Marinova, At. Data and Nucl. Data Tables {bf 99}, 69-95 (2013)].
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