No. The infectious prions make normal prions change their shape to that of infectious prions. But they do not catalyze the synthesis of infectious prions.
The prevailing theory of prion replication involves "autocatalytic protein misfolding." Why wouldn't that fall within your definition of "life"?
Recent research also shows that, although they lack nucleic acids, prions undergo Darwinian evolution, including mutation and natural selection. That finding is irrelevant to your definition, but it would appear to bolster the case for considering prions to be a form of life -- at least in the same sense that viruses could be consider a form of life.