  We present a finite element method for the incompressible Navier--Stokes
problem that is locally conservative, energy-stable and pressure-robust on
time-dependent domains. To achieve this, the space--time formulation of the
Navier--Stokes problem is considered. The space--time domain is partitioned
into space--time slabs which in turn are partitioned into space--time
simplices. A combined discontinuous Galerkin method across space--time slabs,
and space--time hybridized discontinuous Galerkin method within a space--time
slab, results in an approximate velocity field that is $H({\rm
div})$-conforming and exactly divergence-free, even on time-dependent domains.
Numerical examples demonstrate the convergence properties and performance of
the method.
