Context. High angular resolution (sub)millimetre observations of asymptotic giant branch (AGB) stars, now possible with the Atacama Large Millimeter/submillimeter Array (ALMA), allow direct imaging of these objects' photospheres. The physical properties of the molecular material around these regions, which until now has only been studied by imaging of maser emission and spatially unresolved absorption spectroscopy, can be probed with radiative transfer modelling and compared to hydrodynamical model predictions. The prototypical Mira variable, o Cet (Mira), was observed as a Science Verification target in the 2014 ALMA Long Baseline Campaign, offering the first opportunity to study these physical conditions in detail. Aims. With the longest baseline of 15 km, ALMA produces clearly resolved images of the continuum and molecular line emission/absorption at an angular resolution of ∼30 mas at 220 GHz. Models are constructed for Mira's extended atmosphere to investigate the physics and molecular abundances therein. Methods. We imaged the data of 28 SiO = 0, 2 J = 5−4 and H 2 O v 2 = 1 J Ka,Kc = 5 5,0 −6 4,3 transitions and extracted spectra from various lines of sight towards Mira's extended atmosphere. In the course of imaging the emission/absorption, we encountered ambiguities in the resulting images and spectra that appear to be related to the performance of the CLEAN algorithm when applied to a combination of extended emission, and compact emission and absorption. We addressed these issues by a series of tests and simulations. We derived the gas density, kinetic temperature, molecular abundance, and outflow/infall velocities in Mira's extended atmosphere by modelling the SiO and H 2 O lines. Results. We resolve Mira's millimetre continuum emission and our data are consistent with a radio photosphere with a brightness temperature of 2611 ± 51 K. In agreement with recent results obtained with the Very Large Array, we do not confirm the existence of a compact region (<5 mas) of enhanced brightness. Our modelling shows that SiO gas starts to deplete beyond 4 R and at a kinetic temperature of 600 K. The inner dust shells are probably composed of grain types other than pure silicates. During this ALMA observation, Mira's atmosphere generally exhibited infall motion with a shock front of velocity 12 km s −1 outside the radio photosphere. Despite the chaotic nature of Mira's atmosphere, the structures predicted by the hydrodynamical model, codex, can reproduce the observed spectra in astonishing detail, while some other models fail when confronted with the new data. Conclusions. For the first time, millimetre-wavelength molecular absorption against the stellar continuum has been clearly imaged. Combined with radiative transfer modelling, the ALMA data successfully demonstrates the ability to reveal the physical conditions of the extended atmospheres and inner winds of AGB stars in unprecedented detail. Long-term monitoring of oxygen-rich evolved stars will be the key to understanding the unsolved problem of dust ...