The little lamb, who is entirely healthy, happy and running free despite the mutation, was born on a family's farm near Ashbourne, Derbyshire.
Zara Bayley, 10, was helping her father Mark deliver the baby animal when she spotted the extra limb.
She and her brother Jake, 12, have named him after Forrest Gump in the hope he will run free, and persuaded their dad Mark to let them keep him as a pet.
Mark, 47, who is a director of Mark Bayley Ltd, said that he had to look on the internet to make sure he was ok and find out if anything like this had happened before.
"We checked him over and he seems to be fine. He was healthy, drinking from his mother, and doing everything he needed to do."
Forrest seems largely undisturbed by his extra leg, despite not being able to use it properly.
When he moves, the fifth limb appears to dangle at his side rather than move independently.