Even if the universe is infinite, it's impossible to interact or even perceive anything beyond our lightcone. Within that finite (albeit vast) region of space is a finite amount of matter and energy which is constantly and irrevocably subject to entropy, and will eventually be torn apart and decohere with the heat death of the universe.

So... no. In literally no way known to physics can anything other than spacetime itself grow exponentially forever, and even that may not necessarily be the case.