  We report the detection and investigate the properties of high-excitation
lambda-doubling line emission of hydroxyl (OH) detected towards three
asymptotic giant branch (AGB) stars (W Hya, R Dor, and IK Tau) using ALMA. The
OH lines are produced very close to the central stars and seem optically thin
and with no maser effect. We analyse the molecular excitation using population
diagrams and find rotational temperatures of $\sim 2500$ K and column densities
of $\sim 10^{19}$ cm$^{-2}$ for both W Hya and R Dor. For W Hya, we observe
emission from vibrationally excited H2O arising from the same region as the OH
emission. Moreover, CO $v = 1; J = 3 - 2$ emission also shows a brightness peak
in the same region. Considering optically thin emission and the rotational
temperature derived for OH, we find a CO column density $\sim 15$ times higher
than that of OH, within an area of ($92 \times 84$) mas$^2$ centred on the OH
emission peak. These results should be considered tentative because of the
simple methods employed. The observed OH line frequencies differ significantly
from the predicted transition frequencies in the literature, and provide the
possibility of using OH lines observed in AGB stars to improve the accuracy of
the Hamiltonian used for the OH molecule. We predict stronger OH
lambda-doubling lines at millimetre wavelengths than those we detected. These
lines will be a good probe of shocked gas in the extended atmosphere and are
possibly even suitable as probes of the magnetic field in the atmospheres of
close-by AGB stars through the Zeeman effect.
