Have you "erased settings" recently? There is/was an issue where settings storage was like a hard drive that got fragmented (after many many settings changes, perhaps over the course of several firmware versions) and the way to fix it is to reformat it. So try "erase settings", read and follow the instructions carefully about booting and letting it run. Then I am afraid you need to do all of setup again.
If it still will not save settings, then it sounds to me like you have a hardware problem. Do you have another board that you can try, just to prove that you are doing everything correctly? If it works there, but not on the board in question, that makes it way more likely that you have a hardware problem.
Or perhaps someone else, like f5soh might have an idea to try.