The iconic actor said in a statement: "Lucy and I are utterly grief stricken and respectfully request privacy."
Archie was also an actor, and best known for playing Ollie Coulton in the CBBC show, So Awkward.
A photograph of the young actor on his Instagram account
Speaking about the awful loss, CBBC's head of content Cheryl Taylor said: "He will be greatly missed by us all and our deepest condolences go out to his family and friends at this time."
She added that he was "such a talented young actor".
Production company Channel X North also released a statement which read: "He was an incredibly talented performer and his contribution to So Awkward, on and off-screen, will not be forgotten. As well as hilarious, he was a generous, kind-hearted young man who we had the honour to work with on the show for seven years. Our thoughts are with his family and friends at this time."
Nicholas had previously opened up about his son in an interview, saying that he had inherited the "acting gene".
Nicholas and Archie in 2015
John Challis, who played Boycie in Only Fools and Horses, tweeted: "The saddest news of all today. The death of Archie Lyndhurst, just starting out on his chosen career in acting. My heart aches for Nick and Lucy. They could not have loved him more."
Others were quick to offer their condolences, with one writing: "Terrible news, no parent should lose their child."
Another added: "Such a rising talent following in the footsteps of his father, tragically taken from us too soon. My condolences to the cast of So Awkward, Nicholas and his family, and to everyone at CBBC. You will be missed, Archie."
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