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.