Two-terminal memory elements, or memelements, capable of co-locating signal processing and memory via history-dependent reconfigurability at the nanoscale are vital for next-generation computing materials striving to match the brain's efficiency and flexible cognitive capabilities. While memory resistors, or memristors, have been widely reported, other types of memelements remain underexplored or undiscovered. Here we report the first example of a volatile, voltage-controlled memcapacitor in which capacitive memory arises from reversible and hysteretic geometrical changes in a lipid bilayer that mimics the composition and structure of biomembranes. We demonstrate that the nonlinear dynamics and memory are governed by two implicitly-coupled, voltage-dependent state variables-membrane radius and thickness. Further, our system is capable of tuneable signal processing and learning via synapse-like, short-term capacitive plasticity. These findings will accelerate the development of low-energy, biomolecular neuromorphic memelements, which, in turn, could also serve as models to study capacitive memory and signal processing in neuronal membranes.