In this paper, we study the problem of user activity detection and large-scale fading coefficient estimation in a random access wireless uplink with a massive MIMO base station with a large number $M$ of antennas and a large number of wireless single-antenna devices (users). We consider a block fading channel model where the $M$-dimensional channel vector of each user remains constant over a coherence block containing $L$ signal dimensions in time-frequency. In the considered setting, the number of potential users $K_\text{tot}$ is much larger than $L$ but at each time slot only $K_a<<K_\text{tot}$ of them are active. Previous results, based on compressed sensing, require that $K_a\leq L$, which is a bottleneck in massive deployment scenarios such as Internet-of-Things and unsourced random access. In this work we show that such limitation can be overcome when the number of base station antennas $M$ is sufficiently large. We also provide two algorithms. One is based on Non-Negative Least-Squares, for which the above scaling result can be rigorously proved. The other consists of a low-complexity iterative componentwise minimization of the likelihood function of the underlying problem. Finally, we use the discussed approximated ML algorithm as the decoder for the inner code in a concatenated coding scheme for unsourced random access, a grant-free uncoordinated multiple access scheme where all users make use of the same codebook, and the massive MIMO base station must come up with the list of transmitted messages irrespectively of the identity of the transmitters. We show that reliable communication is possible at any $E_b/N_0$ provided that a sufficiently large number of base station antennas is used, and that a sum spectral efficiency in the order of $\mathcal{O}(L\log(L))$ is achievable.