A fluoropolymer optical fiber coating based on the thermoplastic copolymer of chlorotrifluoroethylene and vinylidene fluoride is presented. Such coatings can be used as a UV-transparent material for writing single Bragg gratings or arrays of chirped fiber Bragg gratings directly through the fiber coating with the use of excimer laser radiation at 248 nm. As an optimum radiation density that does not lead to significant degradation of the fluoropolymer coating, an exposure time not exceeding 200 s at 70 mJ/cm2 was identified. With such inscription parameters it was possible to inscribe arrays of fiber Bragg gratings in hydrogen-loaded birefringent optical fiber with an elliptical stress cladding through a 12µm thick coating, so that stripping of the coating is avoided and good mechanical strength is preserved. The reflection spectrum width of the chirped Bragg gratings was about 3.5 nm with a reflectance coefficient of the most effective grating of up to 38%. Such Bragg grating arrays are especially interesting as reflective elements in fiber interferometers.