In application verticals that rely on mission-critical control, such as cooperative intelligent transport systems (C-ITS), 5G-Advanced networks must be able to provide dynamic positioning with accuracy down to the centimeter level. To achieve this level of precision, technology enablers, such as massive multiple-input multiple-output (mMIMO), millimeter waves (mmWave), machine learning and cooperation are of paramount importance. In this paper, we propose a cooperative deep learning (DL)-based positioning methodology that combines these key technologies into a new promising solution for precise 5G positioning. Sparse channel impulse response (CIR) data are used by the positioning infrastructure to extract position-dependent features. We model the problem as a joint task composed of non-line-of-sight (NLOS) identification and position estimation which permits to suitably handle geometrical location measurements and channel fingerprints. The network of base stations (BSs) automatically steers between egocentric (in case of NLOS) and cooperative (for LOS) positioning mode. We perform extensive standard-compliant simulations in a 5G urban micro (UMi) vehicular scenario obtained by ray-tracing and simulation of urban mobility (SUMO) software. Results show that the proposed cooperative DL architecture is able to outperform conventional geometrical positioning algorithms operating in LOS by 47%, achieving a median error of 71 cm on unseen trajectories.