The "Ashfield streets" (Ashfield Court / Crescent / Drive / Gardens) would appear to be named after Ashfield estate which is shown on the OSNI Historical Second Edition (1846 - 1862). "DESIRABLE INVESTMENT. To be Sold by Private CONTRACT, ASHFIELD HOUSE, Offices, Garden, and Grounds, situate on the Antrim and Skegoneill Roads, containing about 5½ acres, as at present in the possession of Mr. Thomas M'Clelland, Solicitor, held for Ever, at the yearly rent of £80. Also the Parcel of LAND adjoining, containing 31 Acres, held in Perpetuity, subject to the yearly rent of £42. Apply to WILLIAM HARTLEY & CO., 16 Waring Street, Belfast,1st March, 1879. (Northern Whig - Tuesday 04 March 1879).
Note the name Ashfield has been used elsewhere in Belfast, e.g. Ashfield House on Ravenhill Road and Ashfield Street off Castlereagh Road, neither of which still exist.