THE BALMY, subs. phr. (common).Sleep: as adj. = sleepy: cf. balmy slumbers (SHAKESPEARE) and balmy sleep (YOUNG). TO HAVE A DOSE (or WINK) OF THE BALMY = to go to sleep. See BEDFORDSHIRE.
1840. DICKENS, The Old Curiosity Shop, ch. viii., p. 42. As its rather late, Ill try and get A WINK OR TWO OF THE BALMY.
See BARMY.