Peter was born in Horsham in the state of Victoria, Australia in 1934. His father Stanley Morris was a civil engineer, and a twice medal winner in the Premier Australian Football League (AFL). His mother, Mary (née Hennessy), was a pharmacist. His father died suddenly at the age of 49 from a heart attack, when Peter was 14, and tragedy hit again a year later when his younger brother, Stan, was killed in a car accident.
At Melbourne University, Peter switched from engineering to medicine and was first introduced to immunology by Sir McFarlane Burnett who later shared the Nobel Prize for Medicine with Sir Peter Medawar. He excelled at sport, representing Australia in University baseball and cricket. He graduated in 1957, started his surgical training in Melbourne, and married Jocelyn. They then travelled to England, working their passage on a cargo ship.
He continued his surgical training in Southampton and was a surgical registrar at the Hammersmith Hospital when the first living non-related kidney transplant was performed.