Jack Mould was driving on Summit Rd near Godley Head park on Wednesday when he found the lamb and took it to the vet to have the arrow removed.
However, he recieved an update from the vet about noon on Thursday saying the lamb "didn't make it."
The SPCA is investigating the incident.
After Mr Mould found the lamb on Wednesday he said on a Facebook post: "If anybody knows anything about this, or especially the people that shot the poor lamb, please can you contact the SPCA or at least message me and I’ll pass it on," he said on Facebook.
"For the people that did this, I hope you feel ashamed and regret your actions. You acted stupidly and carelessly. This is a baby animal got shot and left [to die] injured, bleeding and in a lot of pain. This was a heartless act."
A SPCA spokeswoman said anyone who witnessed the incident or knows further information to lodge an animal welfare complaint.
"Shooting an animal and causing it suffering in this way is an offence under the Animal Welfare Act," the spokeswoman said.
"If SPCA Inspectors were able to gather sufficient evidence to prove a particular person shot an animal in this way, we could be looking at either an 'ill-treatment', 'reckless ill-treatment', or 'wilful ill-treatment' charge, depending on the intent of the person shooting, and the outcome for the animal."
