In this paper, we present a detailed description of a non- parametric two-dimensional (2-D) procedure to extrapolate a signal, denoted Adaptive Weighted Norm Extrapolation (AWNE), and we propose its application for SAR image formation. The benefits of the AWNE procedure are shown when it is applied to the MSTAR targets database of images. Once the phase history is recovered, the AWNE method is applied to a subaperture or to the full set of frequency samples to extrapolate them to a larger aperture. Then, the Inverse DFT is applied to obtain the new complex SAR image. Use of the 2-D AWNE procedure proves to be superior to its one-dimensional version by reducing undesirable effects such as sidelobe interference, and variability in energy of the extrapolated data from row to row and from column to column. To assess the performance of AWNE in enhancing prominent scatterers, reducing speckle, and suppressing clutter, we compare the super-resolved images to the images formed with the traditional Fourier technique starting from the same frequency samples. Both images are also compared with images formed starting from less data to assess the quality of the extrapolation and to quantify the ability to recover from lost resolution. We quantify the performance with the help of a target mask produced by a CFAR detector using metrics such as peak location blob matching count and a mean minimum peak distance. Another focus of our experiments is the illustration of the potential advantages of going beyond the traditional limits of resolution by extrapolating the full aperture of phase history to a larger size. We quantify performance by visual comparison and by the use of a geometric constellation of prominent point scatterers of the targets extracted from the images.