Samuel Willoughby Duffield states that this is a Jesuit hymn, collected by Walraff (1806) from "Psalteriolum Cantionum Catholicarum a Patribus Societatis Jesu."
Further, he says that the hymn anticipates "The Virgin Mary to the Child Jesus" by Mrs. Browning, and "Babe Jesus Lay on Mary's Lap" by Dr. George Macdonald.
"The Latin Hymn-writers and their Hymns," 1889 (Google ebook online).

An article in The Musical Times, vol. 42, 1901, "Music in Scotland: A Brief Historical Survey (Concluded)," claims "Everyone knows the hymn beginning 'Dormi, fili, dormi."
It states that there are nine stanzas.