  \eta\ Car is the only colliding-wind binary for which high-energy \gamma\
rays are detected. Although the physical conditions in the shock region change
on timescales of hours to days, the variability seen at GeV energies is weak
and on significantly longer timescales. The \gamma-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 \eta\ Car. We find that emission from primary
electrons is likely not responsible for the \gamma-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.
