Nah they’re a single molecule. While they do have a mechanism to “reproduce”, they cannot react to stimuli of any kind, or evolve. Of the 7 commonly accepted traits of life, viruses have 5-6 depending on where you stand with them not being able to reproduce on their own. (In comparison, while a tapeworm or other parasite might need a host, they bring their reproductive equipment with them).

Prions have 1 of those traits. They can’t regulate an internal environment as they cannot have one, they lack any kind of organizational trait, they have no metabolism (the other one viruses lack), they do not grow, they don’t adapt to their environment, and they do not respond to stimuli.

A digital thermometer has organization and responds to stimuli, so it’s more alive than a prion.