We report measurements of the electrical current fluctuations generated by peeling a pressure sensitive adhesive tape from a smooth, clean copper substrate in the approximatly 180-degrees peel geometry at speeds between 2 and 6 mm/s. Spectral analysis of these fluctuations using Fourier transform techniques reveals broad bands in the region of 0-150 Hz at these peel speeds. A significant transition in the spectral behavior is observed at peel speeds between 4 and 6 mm/s. The general appearance, spectral behavior, and autocorrelation functions of the fluctuations are all suggestive of chaos. Dimensional analysis of the current signals indicates that the fluctuations are associated with a strange (fractal) attractor of dimension 5.5-5.6. The finite dimension rules out the possibility that these fluctuations are random. The close relation between the current signals and the mechanical events accompanying peeling suggests that the peel event itself is fractal in a spatial-temporal sense, and therefore chaotic in nature.