This paper considers the problem of beam alignment in a cell-free massive MIMO deployment with multiple access points (APs) and multiple user equipments (UEs) simultaneously operating in the same millimeter wave frequency band. Assuming the availability of a control channel at sub-6 GHz frequencies, a protocol is developed that permits estimating, for each UE, the strongest propagation path from each of the surrounding APs, and to perform user-centric association between the UEs and the APs. Estimation of the strongest paths from nearby APs is realized at the UE in a one-phase procedure, during which all the APs simultaneously transmit on pseudo-randomly selected channels with pseudo-random transmit beamformers. An algorithm for orthogonal channels assignment to the APs is also proposed, with the aim of minimizing the mutual interference between APs that transmit on the same channels. The performance of the proposed strategy is evaluated both in terms of probability of correct detection of the directions of arrival and of departure associated to the strongest beam from nearby APs, and in terms of downlink and uplink signal-to-interference-plus-noise ratio. Numerical results show that the proposed approach is effective and capable of efficiently realizing beam alignment in a multi-UE multi-AP wireless scenario.