We investigate the non-orthogonal coexistence between the ultra-reliable low-latency communication (URLLC) and the enhanced mobile broadband (eMBB) in the downlink of a cell-free massive multiple-input multiple-output (MIMO) system. We provide a unified information-theoretic framework that combines a finite-blocklength analysis of the URLLC error probability based on the use of mismatched decoding with an infinite-blocklength analysis of the eMBB spectral efficiency. Superposition coding and three levels of puncturing are considered as alternative downlink coexistence strategies to cope with the inter-service interference and the URLLC random activation pattern, under the assumption of imperfect pilot-based channel state information acquisition at the access points and statistical channel knowledge at the users. Numerical results shed light into the trade-off between eMBB and URLLC performances considering different precoding and power control strategies.