The combination of chalcogenide glasses with polymer photonic crystal fibers (PCFs) is a difficult and challenging task due to their different thermo-mechanical material properties. Here we report the first experimental realization of a hybrid polymer-chalcogenide PCF with integrated As2S3 glass nanofilms at the inner surface of the air-channels of a poly-methyl-methacrylate (PMMA) PCF. The integrated high refractive index glass films introduce distinct antiresonant transmission bands in the 480-900 nm wavelength region. We demonstrate that the ultra-high Kerr nonlinearity of the chalcogenide glass makes the polymer PCF nonlinear and provides a possibility to shift the transmission band edges as much as 17 nm by changing the intensity. The proposed fabrication technique constitutes a new highway towards all-fiber nonlinear tunable devices based on polymer PCFs, which at the moment is not possible with any other fabrication method.