Dunbar was born in 1944 in Scunthorpe, Lincolnshire, and is one of four children. Her father was a steel-worker and her mother was a fishing net maker. She grew up in Lincolnshire.
Dunbar attended Goldsmiths College in London, where she received a Bachelor of Arts in English. After that, she did several jobs, working as a nanny, a waitress, a barmaid, and a salesperson. In 1968, she started working as a teacher in a college drama department of Stratford-on-Avon, England. However, due to her gradual loss of hearing, Dunbar had to stop her teaching career and in 1989, she became a full-time writer.
Dunbar has two grown up children: Ben, a fashion photographer and Polly, an author illustrator. Dunbar currently lives in Norwich.
Dunbar published her first children's book at age 35. In 1985, Dunbar published Mundo and the Weather-Child – a novel about the imaginary friend of a deaf child, which helped her become a runner up for the Guardian Fiction Award. In 1990, her book A Bun for Barney was made into an interactive video game by BBC Multimedia Corporation.
In 1998, she wrote Tell Me Something Happy Before I Go To Sleep, which is recommended as a book to help children feel secure. In 2002 Dunbar did a book tour in the United States to promote this book. Her 2005 picture book Shoe Baby, illustrated by her daughter Polly, was made into a puppet show and is part of the 2006 Brighton Festival.
Dunbar most well-known series, Mouse and Mole, has been adapted into a 26-part television animation series by Grasshopper Productions, with voices lent by Alan Bennett and Richard Briers.
Being a person with a hearing impairment, Dunbar has participated in a number of campaigns on behalf of deaf people. In 1998, Dunbar cycled across Cuba in order to raise funds for the National Deaf Children's Society. Her journal Cycle Cuba, a record of this event, was published in 1999. That same year, she had a trip to the Himalayas in support of the founding of a new ashram. Dunbar has also taught English writing for children from Greek island Skyros.
Dunbar is on the steering group for the in the Picture project run by SCOPE, which is about the representation of children with disability in picture books.