This paper proposes a deep learning-based channel estimation method for multi-cell interference-limited massive MIMO systems, in which base stations equipped with a large number of antennas serve multiple single-antenna users. The proposed estimator employs a specially designed deep neural network (DNN) based on the deep image prior (DIP) network to first denoise the received signal, followed by conventional least-squares (LS) estimation. We analytically prove that our LS-type deep channel estimator can approach minimum mean square error (MMSE) estimator performance for high-dimensional signals, while avoiding complex channel inversions and knowledge of the channel covariance matrix. This analytical result, while asymptotic, is observed in simulations to be operational for just 64 antennas and 64 subcarriers per OFDM symbol. The proposed method also does not require any training and utilizes several orders of magnitude fewer parameters than conventional DNNs. The proposed deep channel estimator is also robust to pilot contamination and can even completely eliminate it under certain conditions.