The young actor was known for his role as Ollie Coulton in the CBBC series So Awkward.
A statement from CBBC confirmed news of his death on Thursday.
It read: “We’re so sorry to have to tell you that Archie Lyndhurst, who so brilliantly played Ollie in So Awkward on CBBC, has very sadly died after a short illness.
“Archie will be hugely missed by all who worked with him, and by all the fans who laughed along with Ollie in the show.”
His father Nicholas later issued a statement, which said: “Lucy and I are utterly grief stricken and respectfully request privacy.”
Speaking to the BBC, CBBC’s head of content Cheryl Taylor described Archie as “such a talented young actor”, saying: “All of us at BBC Children’s are devastated.
“He will be greatly missed by us all and our deepest condolences go out to his family and friends at this time.”
Channel X North, the independent production company that makes So Awkward, also said it was “deeply shocked and saddened” by Archie’s death.
It said in a statement: “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.”
As well as his role on So Awkward, Archie also starred as a young version of Jack Whitehall’s Bad Education character Alfie Wickers in the sitcom in 2014.
Archie’s friend and So Awkward co-star Samuel Small is among those who have paid tribute online.
He said the late actor was “so full of life and had touched so many people’s hearts” in a post on Instagram.
“To my brother Archie. I still can’t quite comprehend that you have passed. You still had so much life to live and I’d give anything for you to keep on living it,” Samuel wrote.
“My heart aches all day and everyday thinking about you and I just wish I could talk to you just one more time so you could know how much I loved you. I vow to make you proud with every last breath. Love you from now until infinity. Rest in power bro.”
He continued: “I wish you could’ve all known Archie how we knew him. I’ve never met someone so full of life and had touched so many peoples hearts. Archie was there for me in my lowest lows but my highest highs were with him. Archie’s my best friend and to say goodbye so soon is absolutely devastating, but his calling was obviously somewhere else and I hope he is brightening up wherever he is.
“To be honest, I still can’t find the words that do him justice and show how much of a beautiful soul he was, but his mission was to make as many people happy as he could and trust me, he really really did.
“Archie loved his friends and family so so dearly and we hope he knows how much he truly was loved. We never know what tomorrow will bring so hug the ones you love extra tight as Archie always did.”
Archie’s mum Lucy Lyndhurst commented: “He will love you and watch over you forever. Love you dearly.”
Fellow CBBC stars and former X Factor: Celebrity contestants Max and Harvey added: “I’m honestly speechless. RIP Archie, sending love to you and his family bro.”
John Challis, who appeared alongside Nicholas in Only Fools And Horses, said his “heart ached” for his former co-star and his wife.
He 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.”