We experimentally study electromagnetically induced transparency cooling of the drumhead modes of planar two-dimensional arrays with up to ions stored in a Penning trap. Substantial sub-Doppler cooling is observed for all drumhead modes. Quantitative measurements for the center-of-mass mode show near ground-state cooling with motional quantum numbers of obtained within . The measured cooling rate is faster than that predicted by single particle theory, consistent with a quantum many-body calculation. For the lower frequency drumhead modes, quantitative temperature measurements are limited by frequency instabilities, but near ground-state cooling of the full bandwidth is strongly suggested. This advance will greatly improve the performance of large trapped ion crystals in quantum information and metrology applications.