η Car is the only colliding-wind binary for which high-energy γ rays are detected. Although the physical conditions in the shock region change on time-scales of hours to days, the variability seen at GeV energies is weak and on significantly longer time-scales. The γ-ray spectrum exhibits two features that can be interpreted as emission from the shocks on either side of the contact discontinuity. Here, we report on the first time-dependent modelling of the non-thermal emission in η Car. We find that emission from primary electrons is likely not responsible for the γ-ray emission, but accelerated protons interacting with the dense wind material can explain the observations. In our model, efficient acceleration is required at both shocks, with the primary side acting as a hadron calorimeter, whilst on the companion side acceleration is limited by the flow time out of the system, resulting in changing acceleration conditions. The system therefore represents a unique laboratory for the exploration of hadronic particle acceleration in non-relativistic shocks.