No, as the recommended distance for children between 3 to 5 years of age is about 0.65 miles. However, this usually depends on the physical condition of the child and other factors such as weather. Also, a 3 year old can complete a mile but he or she will have to take several brakes to cool down and regain energy.
Possibly after alot of training. This is not something a child normally could step out on a track and run 3 miles. I would also check with the pediatrician and get an okay for the training set up before you start this.