"Invariant", a short story by John R. Pierce, first published in Astounding Science Fiction, April 1944, available at the Internet Archive. Does any of these covers ring a bell?
"Dr. Green," I said, "the year is 2170. This is the twenty-second century."
He looked at me baffled, but this time not with disbelief. A strange sort of terror was spread over his features.
"An accident?" he asked. "My memory?"
"There has been no accident," I said. "Your memory is intact, as far as it goes. Listen to me. Concentrate."
Then I told him, simply and briefly, so that his thought processes would not lag. As I spoke to him he stared at me apprehensively, his mind apparently racing. This is what I said:
"Your experiment succeeded, beyond anything you had reason to hope. Your tissues took on the ability to reform themselves in exactly the same pattern year after year. Their form became invariant.
"Photographs and careful measurements show this, from year to year, yes, from century to century. You are just as you were over two hundred years ago.
"Your life has not been devoid of accident. Minor, even major, wounds have left no trace in healing. Your tissues are invariant.
"Your brain is invariant, too; that is, as far as the cell patterns are concerned. A brain may be likened to an electrical network. Memory is the network, the coils and condensers, and their interconnections. Conscious thought is the pattern of voltages across them and currents flowing through them. The pattern is complicated, but transitory—transient. Memory is changing the network of the brain, affecting all subsequent thoughts, or patterns in the network. The network of your brain never changes. It is invariant.
[. . . .]
Green's head had sunk to his chest. His face was troubled, and he seemed to seek solace in the warmth of the fire. The dog at his feet stirred, and he looked down, a sudden smile on his face. I knew that his train of thought had been interrupted. The transients had died from his brain. Our whole meeting was gone from his pattern of thought.