Power was born in Eastbourne, East Sussex. She began singing and dancing lessons at the age of four, prompted by her mother, who was very keen for her to perform. However, at the age of 14, Power concluded "that she wasn't physically suited to be a dancer", so she decided to take up acting on the advice of her teacher. Power auditioned for a place at the Royal Academy of Dramatic Art (RADA) at the age of 17, and was accepted. She has commented "I just glided into it so I was still naive…Rada was good and I realised there was more than just going on stage and being clapped." Power graduated from RADA in 1966.