Mayhew died in his north Texas home surrounded by family, according to the statement.
They did not confirm his illness but TMZ reported his death was due to a heart condition.
The London-born actor played Chewbacca in Star Wars (1977), The Empire Strikes Back (1980), Return of the Jedi (1983), Revenge of the Sith (2005) and The Force Awakens (2015).
He had been working at a hospital when he tried out for the part that would make him famous - director George Lucas had been looking for someone taller than Darth Vadar (who was 6ft 6in). Mayhew stood 7ft 3in.
Mayhew did not provide the voice for the Wookiee but he was credited for the body language and expressions that made the hairy alien a fan favourite.
"He put his heart and soul into the role of Chewbacca and it showed in every frame of the films from his knock kneed running, firing his bowcaster (a weapon in the films) from the hip, his bright blue eyes, down to each subtle movement of his head and mouth," his family said.
"But to him, the Star Wars family meant so much more to him than a role in a film.
"The relationships that began then grew into the friends and family that he would love for decades to come.
"As the films released and became more well-known, he had his eyes opened to the possibilities of what he could achieve with his role in the Star Wars universe."
These possibilities included a number of charities and organisations - Wounded Warriors, Make A Wish, the 501st Legion.
He also established the Peter Mayhew Foundation which supported a wide range of needs - from families in crisis to food supplies for children in Venezuela.
He became a US citizen in 2005 and is survived by his wife Angie and three children.
Angie will take control of his foundation, where his "kindness, generosity and care will live on", the family said.
A memorial service for friends and family will be held on 29 June and a public memorial in early December in Los Angeles.
© Sky News 2019