The story would be set in a sort of medieval period, and the character in present day would be about 16 years old.
John Hopkins Hospital's website answer's your question pretty well:
Third-degree burns destroy the epidermis and dermis. Third-degree burns may also damage the underlying bones, muscles, and tendons. When bones, muscles, or tendons are also burned, this may be referred to as a fourth-degree burn. The burn site appears white or charred. There is no feeling in the area since the nerve endings are destroyed... Persons with severe burns may be left with a loss of certain physical abilities, including loss of limb(s), disfigurement, loss of mobility, scarring, and recurrent infections because the burned skin has decreased ability to fight infection. In addition, severe burns can penetrate deep skin layers, causing muscle or tissue damage that may affect every system of the body.
That is with modern medicine. With medieval medicine, they would be lucky to not die from infections.
I'm gonna partially disagree with the answer from John Hopkins - because that answer is targeted at an Adult.
Babies, on account of them still growing, can withstand and recover from injuries that would otherwise cause life-long disabilities in Adult.
Assuming that the infant doesn't die from Infection (which as the other answer pointed out is the most likely scenario) - then with a few caveats - I'm gonna say it's possible they would retain the majority of their mobility in that arm as they grow and the damaged tissue is replaced.
I don't think so. As long as bones, muscles, and tendons are intact, and infections can be treated, losing mobility is not inevitable.
Third-degree burns destroy the epidermis and dermis. Third-degree burns may also damage the underlying bones, muscles, and tendons.
source (my emphasis)
Infections have been successfully treated since centuries. While it wasn't always clear why treatments worked, knowledge and experiences were shared.
Among others, honey, sea water, alcoholic drinks, and onions have been used to "disinfect" and clean the skin in ancient and medieval times. E.g.:
The burn was prevented from becoming dry by using anointments placed on the burn. These anointments typically consisted of vinegar, egg, rose oil, opium, and a multitude of different herbs. The ointment was applied to affected area, and then reapplied as needed.