Out-of-equilibrium self-assembly of proteins such as actin and tubulin is a key regulatory process controlling cell shape, motion and division. The design of functional nanosystems based on dissipative self-assembly has proven to be remarkably difficult due to a complete lack of control over the spatial and temporal characteristics of the assembly process. Here, we show the dissipative self-assembly of FtsZ protein (a bacterial homologue of tubulin) within coacervate droplets. More specifically, we show how such barrier-free compartments govern the local availability of the energy-rich building block guanosine triphosphate, yielding highly dynamic fibrils. The increased flux of FtsZ monomers at the tips of the fibrils results in localized FtsZ assembly, elongation of the coacervate compartments, followed by division of the fibrils into two. We rationalize the directional growth and division of the fibrils using dissipative reaction-diffusion kinetics and capillary action of the filaments as main inputs. The principle presented here, in which open compartments are used to modulate the rates of dissipative self-assembly by restricting the absorption of energy from the environment, may provide a general route to dissipatively adapting nanosystems exhibiting life-like behaviour.