Cell-free Massive multiple-input multiple-output (MIMO) ensures ubiquitous communication at high spectral efficiency (SE) thanks to increased macro-diversity as compared cellular communications. However, system scalability and performance are limited by fronthauling traffic and interference. Unlike conventional precoding schemes that only suppress intracell interference, full-pilot zero-forcing (fpZF), introduced in [1], actively suppresses also inter-cell interference, without sharing channel state information (CSI) among the access points (APs). In this study, we derive a new closed-form expression for the downlink (DL) SE of a cell-free Massive MIMO system with multi-antenna APs and fpZF precoding, under imperfect CSI and pilot contamination. The analysis also includes max-min fairness DL power optimization. Numerical results show that fpZF significantly outperforms maximum ratio transmission scheme, without increasing the fronthauling overhead, as long as the system is sufficiently distributed.