Traditional beamforming of medical ultrasound images relies on sampling rates significantly higher than the actual Nyquist rate of the received signals. This results in large amounts of data to store and process, imposing hardware and software challenges on the development of ultrasound machinery and algorithms, and impacting the resulting performance. In light of the capabilities demonstrated by deep learning methods over the past years across a variety of fields, including medical imaging, it is natural to consider their ability to recover high-quality ultrasound images from partial data. Here, we propose an approach for deep-learning-based reconstruction of B-mode images from temporally and spatially sub-sampled channel data. We begin by considering sub-Nyquist sampled data, time-aligned in the frequency domain and transformed back to the time domain. The data are further sampled spatially so that only a subset of the received signals is acquired. The partial data is used to train an encoder-decoder convolutional neural network (CNN), using as targets minimum-variance (MV) beamformed signals that were generated from the original, fully-sampled data. Our approach yields high-quality B-mode images, with up to two times higher resolution than previously proposed reconstruction approaches (NESTA) from compressed data as well as delay-and-sum (DAS) beamforming of the fully-sampled data. In terms of contrast-to- noise ratio (CNR), our results are comparable to MV beamforming of the fully-sampled data, and provide up to 2 dB higher CNR values than DAS and NESTA, thus enabling better and more efficient imaging than what is used in clinical practice today.