We report extensive observational data for five of the lowest redshift Super-Luminous Type Ic Supernovae (SL-SNe Ic) discovered to date, namely, PTF10hgi, SN2011ke, PTF11rks, SN2011kf, and SN2012il. Photometric imaging of the transients at +50 to +230 days after peak combined with host galaxy subtraction reveals a luminous tail phase for four of these SL-SNe. A high-resolution, optical, and near-infrared spectrum from xshooter provides detection of a broad He I lambda 10830 emission line in the spectrum (+50 days) of SN2012il, revealing that at least some SL-SNe Ic are not completely helium-free. At first sight, the tail luminosity decline rates that we measure are consistent with the radioactive decay of Co-56, and would require 1-4 M-circle dot of Ni-56 to produce the luminosity. These Ni-56 masses cannot be made consistent with the short diffusion times at peak, and indeed are insufficient to power the peak luminosity. We instead favor energy deposition by newborn magnetars as the power source for these objects. A semi-analytical diffusion model with energy input from the spin-down of a magnetar reproduces the extensive light curve data well. The model predictions of ejecta velocities and temperatures which are required are in reasonable agreement with those determined from our observations. We derive magnetar energies of 0.4 less than or similar to E(10(51) erg) less than or similar to 6.9 and ejecta masses of 2.3 less than or similar to M-ej(M-circle dot) less than or similar to 8.6. The sample of five SL-SNe Ic presented here, combined with SN 2010gx-the best sampled SL-SNe Ic so far-points toward an explosion driven by a magnetar as a viable explanation for all SL-SNe Ic.