  We introduce a two-dimensional generalisation of the quasiperiodic
Aubry-Andr\'e model. Even though this model exhibits the same duality relation
as the one-dimensional version, its localisation properties are found to be
substantially more complex. In particular, partially extended single-particle
states appear for arbitrarily strong quasiperiodic modulation. They are
concentrated on a network of low-disorder lattice lines, while the rest of the
lattice hosts localised states. This spatial separation protects the localised
states from delocalisation, so no mobility edge emerges in the spectrum.
Instead, localised and partially extended states are interspersed, giving rise
to an unusual type of mixed spectrum and enabling complex dynamics even in the
absence of interactions. A striking example is ballistic transport across the
low-disorder lines while the rest of the system remains localised. This
behaviour is robust against disorder and other weak perturbations. Our model is
thus directly amenable to experimental studies and promises fascinating
many-body localisation properties.
