In this work we propose a technique to remove sparse impulse noise from hyperspectral images. Our algorithm accounts for the spatial redundancy and spectral correlation of such images. The proposed method is based on the recently introduced Blind Compressed Sensing (BCS) framework, i.e. it empirically learns the spatial and spectral sparsifying dictionaries while denoising the images. The BCS framework differs from existing CS techniques - which assume the sparsifying dictionaries to be data independent, and from prior dictionary learning studies which learn the dictionary in an offline training phase. Our proposed formulation have shown over 5 dB improvement in PSNR over other techniques.