adj

• (obsolete) Meagre, paltry, scant in quantity.
  "1659, Samuel Cradock, Knowledge & Practice, Or, A Plain Discourse of the Chief Things Necessary to be Known, Believ’d & Practised in order to Salvation, London: John Rothwell, Chapter 17, Of the Duties of the Rich, pp. 494-495, We should think him a very imprudent Husbandman, that to save a little seed at present, would sow so thin, as to spoil his crop. And the same folly ’twill be in us, if by the sparingness and niggardize of our Almes, we make our selves a lank Harvest hereafter, and lose the reward God hath provided for the liberal Almes-giver."
• (of hair) Straight and flat; thin and limp. (Often associated with being greasy.)
  "1695, John Stevens (translator), The Portugues Asia; or, The History of the Discovery and Conquest of India by the Portugues, by Manuel de Faria e Sousa, London: C. Brome, Chapter 10, p. 291, The Inhabitants most simple, and treated them with great affection. Of Colour more inclined to white, of Body strong and comly, lank Hair, and long Beards, their Cloaths of very fine Mats […]"
• (obsolete) Languid; drooping, slack.
  "The water-nymphs, that in the bottom played, Held up their pearled wrists, and took her in, Bearing her straight to aged Nereus’ hall; Who, piteous of her woes, reared her lank head, And gave her to his daughters to imbathe […]"

verb

• (rare, intransitive) To become lank.
  "[…] on the Alps It is reported thou didst eat strange flesh, Which some did die to look on: and all this— It wounds thine honour that I speak it now— Was borne so like a soldier, that thy cheek So much as lank’d not."