Can you die from frostbite if you don't get an amputation?

Yes. Cold exposure that's severe enough to cause frostbite can also cause hypothermia. When your body temperature drops, your heart, nervous system and other organs don't work correctly. Left untreated, hypothermia eventually leads to complete failure of your heart and respiratory system and to death.
It is possible. If the frostbite site becomes severely infected there is a possibility of death if the infection becomes systemic.