We address the origin of ultra-diffuse galaxies (UDGs), which have stellar masses typical of dwarf galaxies but effective radii of Milky Way-sized objects. Their formation mechanism, and whether they are failed L* galaxies or diffuse dwarfs, are challenging issues. Using zoomin cosmological simulations from the Numerical Investigation of a Hundred Astrophysical Objects (NIHAO) project, we show that UDG analogues form naturally in dwarf-sized haloes due to episodes of gas outflows associated with star formation. The simulated UDGs live in isolated haloes of masses 10^10-11 M⊙, have stellar masses of 10^7-8.5 M⊙, effective radii larger than 1 kpc and dark matter cores. They show a broad range of colours, an average Sérsic index of 0.83, a typical distribution of halo spin and concentration, and a non-negligible HI gas mass of 10^7-9 M⊙, which correlates with the extent of the galaxy. Gas availability is crucial to the internal processes which form UDGs: feedback-driven gas outflows, and subsequent dark matter and stellar expansion, are the key to reproduce faint, yet unusually extended, galaxies. This scenario implies that UDGs represent a dwarf population of low surface brightness galaxies and should exist in the field. The largest isolated UDGs should contain more HI gas than less extended dwarfs of similar M*.