Disclaimer: this might be from a German author, at least I read it in German
Read this in the 90s, probably quite older than that.
I'll try to tell the rough story, obviously spoilers towards the end.
It begins with the protagonist in the present/near future, who is some sort of cult leader or preacher, giving a public speech. Suddenly, he turns into an indestructible statue, only turning back after several dozen years. Without realizing it, he somehow discovered a way to do this, and by the time he wakes up, he's a full-blown icon of a growing followership, as others managed to recreate this process for shorter periods.
Sadly, I can't remember how this process is supposed to work or would be consciously initiated in-universe.
My memory of the middle part of the story is (even more) shady, he reconnects with his then-girlfriend, and together they travel further and further into the future, in increasing intervals.
Eventually, they awake in a sealed glass dome, the last bastion of mankind after an alien life form called "wild life" (or similar, again, I read this in German) absorbed all unprotected biomass on Earth. Kinda like the titular "Jam" by Yahtzee Croshaw, although I think it was rather described as a conglomerate of organisms that consumed each other and everything else constantly.
After some more jumps, millions or billions of years into the future, the two and maybe some others are the last living humans, the dome being maintained by (also alien, but friendly) robots. As the dome finally collapses and lets in a flood of wild life, they dare the "big jump" beyond the end of time. The story ends there, but it is implied they succeed.