JHKs photometry is presented from a 3-yr survey of the central regions of the Local Group dwarf irregular galaxy IC 1613. The morphologies of the colour-magnitude and colour-colour diagrams are discussed with particular reference to the supergiants and M- and C-type asymptotic giant branch (AGB) stars. Mean JHKs magnitudes, amplitudes and periods are given for five O-rich and nine C-rich Mira variables for which bolometric magnitudes are also estimated. A distance of 750kpc ((m-M)0=24.37+/-0.08mag) is derived for IC 1613 by fitting a period-luminosity (PL) relation to the C-rich Miras. This is in agreement with values from the literature. The AGB stars exhibit a range of ages. A comparison with theoretical isochrones suggests that four luminous O-rich Miras are as young as 2x10^8^yr. One of these has a lithium absorption line in its spectrum, demonstrating that it is undergoing hot bottom burning (HBB). This supports the idea that HBB is the cause of the high luminosity of these AGB stars, which puts them above the fundamental PL relation. Further studies of similar stars, selected from their positions in the PL diagram, could provide insight into HBB. A much fainter, presumed O-rich, Mira is similar to those found in Galactic globular clusters. The C Miras are of intermediate age. The O-rich variables are not all recognized as O-rich, or even as AGB stars, on the basis of their J-Ks colour. It is important to appreciate this when using near-infrared surveys to classify AGB stars in more distant galaxies.