Fishwick was born near Accrington, Lancashire. He attended Liverpool School of Art for two years before entering the Naval service in 1942. After WWII he returned to complete his studies and acquire a teacher's diploma (1947). Though his teaching career was ever present, and he became principal of the Exeter College of Art in 1958, retiring in 1984, he always kept up an active painting and exhibition schedule. Particular friends in Cornwall were Michael CANNEY and his wife Madeleine, Jack PENDER, Paul FEILER and Alexander MACKENZIE.

From 1952 to 1983 he and his wife, Patricia FISHWICK, also an artist and a teacher of history of art, were members of the NSA. Clifford also exhibited with the Penwith Society in the 1950s and 60s. In the arts review cutting referenced in Hardie (1995, p114) foremost artists are mentioned as Patrick HERON, Paul FEILER, Peter LANYON, John TUNNARD, Dod PROCTER, Alethea GARSTIN, John WELLS, Bernard LEACH and Clifford Fishwick as all being well-represented, 'together with many youthful painters of promise working in the area.'

The artist-couple lived in Topsham, Devon, nr Exeter, where they both taught.