  The prototypical Mira variable, $o$ Cet (Mira), has been observed as a
Science Verification target in the 2014 ALMA Long Baseline Campaign with a
longest baseline of 15 km. ALMA clearly resolves the images of the continuum
and molecular line emission/absorption at an angular resolution of ~30 mas at
220 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 from
various lines-of-sight towards Mira's extended atmosphere. In the course of
imaging, we encountered ambiguities in the resulting images and spectra that
appear to be related to the performance of the CLEAN algorithm. We resolve
Mira's millimetre continuum emission and our data are consistent with a radio
photosphere with a brightness temperature of 2611$\pm$51 K, in agreement with
recent results obtained with the VLA. We do not confirm the existence of a
compact region (<5 mas) of enhanced brightness. We derive the gas density,
kinetic temperature, molecular abundance and outflow/infall velocities in
Mira's extended atmosphere by modelling the SiO and H$_2$O lines. We find that
SiO-bearing gas starts to deplete beyond 4$R_\star$ and at a kinetic
temperature of $\lesssim$600 K. The inner dust shells are probably composed of
grain types other than pure silicates. During this observation, Mira's
atmosphere generally exhibited infall motion, with a shock front of velocity
$\lesssim$12 km/s outside the radio photosphere. 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. Combined with radiative transfer modelling, ALMA successfully
demonstrates the ability to reveal the physical conditions of the extended
atmospheres and inner winds of AGB stars in unprecedented detail. (Abbreviated
abstract)
