I am surprised whenever I find out that a landlord does an oral agreement. I have heard that if a tenant lives at a location long enough, he or she may have some entitlement to living there that builds up over time. Yikes!
You are referring to adverse possession laws, and the fact that there is an agreement between the owner and the tenant (whether written or verbal) nullifies any claim to adverse possession, which must, by definition, be adverse. If owner agrees, there is no adversity, therefore no possession which could lead to ownership.