Fluid antenna multiple access (FAMA) exploits the spatial opportunities in wireless channels to overcome multiuser interference by position (a.k.a.~port) switching, which can achieve better performance compared to traditional fixed multiple-input multiple-output (MIMO) systems. Additionally, integrated data and energy transfer (IDET) is capable of providing both wireless data transfer (WDT) and wireless energy transfer (WET) services towards low-power devices. In this paper, a FAMA-assisted IDET system is investigated, where a base station (BS) equipped with $N$ fixed antennas provides dedicated IDET services towards $N$ user equipments (UEs). Each UE is equipped with a single fluid antenna, while the power splitting (PS) approach is conceived for coordinating WDT and WET. The outage probabilities of both WDT and WET are derived and approximated into closed-forms, where the fluid antenna (FA) at each UE selects the optimal port to achieve the maximum signal-to-interference-plus-noise ratio (SINR) or the energy harvesting power (EHP). The IDET outage probabilities are defined and subsequently derived and approximated into closed-forms. Further, multiplexing gains of the proposed system are defined and analyzed to evaluate the performace. Numerical results validate the theoretical analysis, while also illustrate that the trade-off is achieved between WDT and WET performance by exploiting different port selection strategies. Furthermore, the number of UEs should be optimized to achieve better IDET performance of the system.