no, it's definitely a public highway according to the local highways authority:-

MIDDLE WICK / LOWER WICK (UNMETALLED)
Road Name: MIDDLE WICK / LOWER WICK (UNMETALLED)

Road Number: 02103978

Road Class: T - Unclass Adopted Unmetalled

Road Reference: 71T29 AL 05

Road District: Basingstoke and Deane