In a post on Twitter, the children's TV channel said all of its team had been left devastated by the death of "such a talented young actor".
Archie was best known for playing Ollie Coulton in the CBBC comedy So Awkward, having starred in the show since it launched in 2015.
He died after a short illness, the CBBC statement said.
"He will be greatly missed by us all and our deepest condolences go out to his family and friends at this time."
The channel also urged any young viewers affected by the news to share their feelings and talk to someone.
Nicholas Lyndhurst said in a statement: "Lucy and I are utterly grief stricken and respectfully request privacy."
Archie's father famously played Rodney Trotter, the hapless younger brother of Sir David Jason's Del Boy, in Only Fools and Horses.
John Challis, who played Aubrey "Boycie" Boyce in the long-running BBC comedy said Archie's death was "the saddest news of all", adding that his "heart aches for Nick and Lucy", his parents.
Archie's friend and So Awkward co-star Samuel Small, who also appeared in Game Of Thrones and BBC drama The Nest, shared several photographs and said he had never known anyone who "had touched so many people's hearts" in an emotional tribute on Instagram.
In response to the tribute, a message from Lucy Lyndhurst's account said: "He will love you and watch over you forever."
Archie started his career at the Sylvia Young theatre school, according to his CV.
As well as CBBC, he also appeared in hospital drama Casualty and in Jack Whitehall comedy Bad Education, as well as other projects playing a young version of the comedian.
His death is not the first tragedy to hit CBBC in recent years.
Sixteen-year-old Millie Inbetween and Almost Never star Mya-Lecia Naylor died in hospital in 2019.