It is possible in certain instances and it depends on the province. It seems to be getting very difficult in the more advanced provinces such as Guangdong, Jiangsu etc and the big tier one cities.

In Shanxi it is possible but if you are over 60 they make you sign a disclaimer that you have your own insurance and will absolve them of any responsibility if anything happens to you. The local university to me has a teacher who is 67 and he was able to renew his resident permit without issue subject to this. Maybe if you could ask for the contact details of another teacher in a similar situation to hear their story.