Origin apparently English, 17th century See: Galway City for a start and chick on the English version as well.
The two versions in the DT are both of the later Irish song. See thread 44796 for more on Irish versions Irish-Spanish
Mention of the English version, dating to 1624, at 11158: Spanish Lady (ies)

The Bodleian Library has about ten broadsides of the older English version, at least one dating to the 17th century. Put Spanish Lady in Search.

Can't find the English original in Supersearch. It is a rather long ballad.