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The prototypical Mira variable, $o$ Cet (Mira), has been observed as aScience Verification target in the 2014 ALMA Long Baseline Campaign with alongest baseline of 15 km. ALMA clearly resolves the images of the continuumand molecular line emission/absorption at an angular resolution of ~30 mas at220 GHz. We image the data of the $^{28}$SiO v=0, 2 $J$=5-4 and H$_2$O$\nu_2$=1 $J(K_a,K_c)$=5(5,0)-6(4,3) transitions and extract spectra fromvarious lines-of-sight towards Mira's extended atmosphere. In the course ofimaging, we encountered ambiguities in the resulting images and spectra thatappear to be related to the performance of the CLEAN algorithm. We resolveMira's millimetre continuum emission and our data are consistent with a radiophotosphere with a brightness temperature of 2611$\pm$51 K, in agreement withrecent results obtained with the VLA. We do not confirm the existence of acompact region (<5 mas) of enhanced brightness. We derive the gas density,kinetic temperature, molecular abundance and outflow/infall velocities inMira's extended atmosphere by modelling the SiO and H$_2$O lines. We find thatSiO-bearing gas starts to deplete beyond 4$R_\star$ and at a kinetictemperature of $\lesssim$600 K. The inner dust shells are probably composed ofgrain types other than pure silicates. During this observation, Mira'satmosphere generally exhibited infall motion, with a shock front of velocity$\lesssim$12 km/s outside the radio photosphere. The structures predicted bythe hydrodynamical model CODEX can reproduce the observed spectra inastonishing detail; while some other models fail when confronted with the newdata. Combined with radiative transfer modelling, ALMA successfullydemonstrates the ability to reveal the physical conditions of the extendedatmospheres and inner winds of AGB stars in unprecedented detail. (Abbreviatedabstract)

Resolving the extended atmosphere and the inner wind of Mira ($o$ Ceti) with long ALMA baselines