Hemingway was born in Minster, Kent and after gaining a diploma from Rochester Technical College he emigrated to Canada in 1955. Serving briefly in the Alberta Department of Public Works, Hemingway founded his own practice in 1956. Hemingway won the Massey medal twice for his architecture: the Peter Hemingway Fitness and Leisure Centre formerly known as Coronation Pool (1968–1970) and the Stanley Engineering Building (1968). However, his most well-known and recognizable building is probably the Muttart Conservatory, a group of pyramid-shaped greenhouses in the Edmonton river valley.  He was also the architect of the Central Pentecostal Tabernacle in Central Edmonton which was demolished in 2007 despite attempts for it to be given protective status. In 1982 he served as President of the Alberta Association of Architects. He was a frequent contributor to professional publications.