Adult neurogenesis or brain plasticity is a fact. Skin cells, blood cells and all types of cells have a finite lifespan - a few days to a few months.
Osteocytes in human bone can live up to 50 years.
"New neurons are made in just two parts of the brain—the hippocampus, involved in memory and navigation, and the olfactory bulb, involved in smell (and even then only until 18 months of age). Aside from that, your neurons are as old as you are and will last you for the rest of your life. They don’t divide, and there’s no turnover."