This is a short story possibly having appeared in one of Isaac Asimov's Great SF Stories books. I have lots of tentative keywords but wasn't able to locate it through Google.
The story revolves around a man with a suitcase; it becomes quickly clear that the suitcase contains a memory engram storage system and the man's memory is in the suitcase. Everyone has a suitcase. Memories get dumped on tape, and can be restored.
And whenever a memory is not dumped and restored, it is quickly lost.
As the story goes on a scenario emerges - every human being has somehow acquired the power of complete, flawless regeneration. We're in the far future, and everyone is immortal - and amnesiac. Hence the suitcases; stealing someone's suitcase is equated to murder.
The main character goes around North America and the flaws of this situation become apparent. No biological change is possible anymore. The "effect" appears to have turned on at some point, one July (or perhaps June) twenty-seventh, and every living system is forever frozen at that date; pregnant women still can't give birth after centuries, newborns do not grow, sores and wounds that were festering on the Day still do not heal.
At some point, the main character inadvertently runs over a dog, killing it. It's a boy's dog, the boy is devastated, and his parents explain to the main character that they're going to remove the memories of ever having had a dog from the boy's suitcase that night.
At this point we understand that this guy is the one responsible for the effect in the first place. There's a transmitter somewhere that saturates the whole planet with regenerating radiation, and he built it and the (next to) inestinguishable power source that powers it.
The guy now invents a screen to allow biological time to go forward, and inside the shielded room gets a bitch pregnant and a puppy born to replace the dog he killed. Then he delivers the puppy to the no longer bereaved family, and they turn the gift down - the man says something like "You know, sometimes I sort of feel as if we had a dog. Crazy feeling.".
In the end, the guy decides to undo his work.
Does anyone know this story?