we recently purchased a cabin built in the 70's. The shower/tub contains a single faucet which keeps dripping for a while after shut off.

I tried to repair the valve by taking off the cover plate that secures the valve to the tub enclosure wall. Inside there is a dome shaped cylinder with a black band which looks like rubber. I could not remove the handle. There was no screw under the plastic circle containing hot and cold indications. I could not find a set screw on the handle. I don't know the manufacturer.
I suspect that this is a ball type faucet.

Any advise on how to repair this faucet will be appreciated.