Gunning seems to have been a rare name in eleventh-century England. No instances other than the small Cheshire landowner discussed here as Gunning 2 are recorded in PASE, and the name does occur not in the early records of personal names and place-names in Yorkshire and Lincolnshire (Fellows Jensen 1968), though one man called Gunning was commemorated in the Durham Liber Vitae (DVL, 18).
Von Feilitzen discussed the Domesday name under the head-form Gunningr, but in fact the name seems not to be recorded in ON records, and it is more likely that it was coined in England by adding the fairly common late Anglo-Saxon personal-name suffix –ing to the Scandinavian name Gunni or to the first element Gunn– (Björkman 1910: 56).
Björkman 1910: Erik Björkman, Nordische Personennamen in England in alt- und frühmittel-englischer Zeit, Studien zur englischen Philologie 37 (Halle an der Saale, 1910; reprinted Tübingen: Max Niemeyer, 1973)
DLV: The Durham Liber Vitae. London, British Library, MS Cotton Domitian A.VII: Edition and Digital Facsimile with Introduction, Codicological, Prosopographical and Linguistic Commentary, and Indexes, ed. D. W. Rollason and L. Rollason, 3 vols. (London: British Library, 2007)
Spellings in Domesday Book: Gunninc
Forms in modern scholarship:
von Feilitzen head forms: Gunningr
Phillimore edition: Gunning
Alecto edition: Gunningr