Stacked intelligent metasurfaces (SIMs) have recently emerged as an effective solution for next-generation wireless networks. A SIM comprises multiple metasurface layers that enable signal processing directly in the wave domain. Moreover, recent advances in flexible metamaterials have highlighted the potential of flexible intelligent metasurfaces (FIMs), which can be physically morphed to enhance communication performance. In this paper, we propose a stacked flexible intelligent metasurface (SFIM)-based communication system for the first time, where each metasurface layer is deformable to improve the system's performance. We first present the system model, including the transmit and receive signal models as well as the channel model, and then formulate an optimization problem to maximize the system sum rate under constraints on the transmit power budget, morphing distance, and the unit-modulus condition of the meta-atom responses. To solve this problem, we develop an alternating optimization framework based on the gradient projection method. Simulation results demonstrate that the proposed SFIM-based system achieves significant performance gains compared to its rigid SIM counterpart.