Fluid antenna enables position reconfigurability that gives transceiver access to a high-resolution spatial signal and the ability to avoid interference through the ups and downs of fading channels. Previous studies investigated this fluid antenna multiple access (FAMA) approach in a single-cell setup only. In this paper, we consider a cell-free network architecture in which users are associated with the nearest base stations (BSs) and all users share the same physical channel. Each BS has multiple fixed antennas that employ maximum ratio transmission (MRT) to beam to its associated users while each user relies on its fluid antenna system (FAS) on one radio frequency (RF) chain to overcome the inter-user interference. Our aim is to analyze the outage probability performance of such cell-free FAMA network when both large- and small-scale fading effects are considered. To do so, we derive the distribution of the received \textcolor{black}{magnitude} for a typical user and then the interference distribution under both fast and slow port switching techniques. The outage probability is finally obtained in integral form in each case. Numerical results demonstrate that in an interference-limited situation, although fast port switching is typically understood as the superior method for FAMA, slow port switching emerges as a more effective solution when there is a large antenna array at the BS. Moreover, it is revealed that FAS at each user can serve to greatly reduce the burden of BS in terms of both antenna costs and CSI estimation overhead, thereby enhancing the scalability of cell-free networks.