According to Oxford English Dictionary, the answer is yes. Zinke was the earliest attested spelling of the word, with zink also preceding zinc.
The earliest attested use of the word is from 1651.
Any sulphurous, and imperfect metall, as Iron, Copper, or Zinke.
- John French · The art of distillation; or, A treatise of the choisest spagyricall preparations · 1st edition, 1651 (1 vol.).
The spelling you suggested is believed from 1734.
We took six Ounces of Copper, and melting it in a Wind-Furnance, added to it an Ounce of Zink.
- Peter Shaw · Chemical Lectures publickly read at London in … 1731, and 1732; and since at Scarborough, in 1733, for the improvement of arts, trades, and natural philosophy · 1734.
And finally, the earliest attested spelling with a 'c' is from 1813
Zinc is one of the most combustible of the common metals.
- Humphry Davy · Elements of agricultural chemistry, in a course of lectures of the Board of Agriculture
You ask: When might it have ceased to be used?
OED indicates that zink was used into the 1800's, in its remarks on the forms of the word:
Forms: 16–18 zink, (16 zinke, 16–17 zinck), 17– zinc.
However, jlovegren proved that some uses from the 1900's can be found. I would add to that these snippets from 1906 and 1914, respectively. However, the dates provided by OED are probably an accurate measure of when each form was in common use.