Conserved environments or sanctuaries were created so that they would protect certain species of animals and try to help repopulate them over time. However, not all conserved environments can protect these animals. It really depends upon why the animal went extinct. The black rhinos are extinct and there were several different sanctuaries for rhinos in the area.
However, they became extinct because they may have been dying off faster than reproducing. In a conserved environment like a sanctuary, the rhinos would be together, but there would be more rhinos in the wild. Therefore, this way of helping them would stop poachers from killing then but may also cause them less likely to reproduce. However, sanctuaries should continue because this is probably a rare situation.
In short, yes animals have gone extinct while in protected areas. Animals and species are going extinct every day at an alarming rate. Sometimes, their extinction can be blamed on the human population as a whole because of the contribution to global warming, modern industry, and deforestation. Where the real problem lies and how these critically endangered animals are going extinct is in poaching and illegal hunting.
Unfortunately, some of these areas are in remote places and hard to monitor by scientists. The remote areas combined with the fact that many places in the world have extremely low wages but offer great pay to those willing to hunt illegally means endangered animals become extinct.