Search:

Banjoman

This is a cut and paste from the wiki, and may be pertinent to your strop.

Stiffness or wrinkles
Some physical influences, such as dust, dirt, or longer periods of non-use, may cause...

I'm not, at all, familiar with their strops. Can you give us a desciption of it including what width it is, and why you think it might need oil? I think we can come up with some solutions. :D
...