Yes, although I don't know if all states have them. In NC they are called Company Police which are considered Special Police Officers commissioned by the Attorney General. It started in the 1800s with the textile mills and company towns. They may only exercise jurisdiction on their employers property, or they may be employees of a security company that is hired and granted jurisdiction by other businesses. Most companies that do it have both security (Private Protective Services) and police (Company Police) services available. The only authority they have outside real property owned by their employer/client is when in hot pursuit.

Private universities, same as public, may have Campus Police instead of Company Police, and those officers also have jurisdiction on public ways passing through the campus. In some cases they also have one mile extraterritorial jurisdiction; moreover, they can receive broader jurisdiction through agreements with the city/county law enforcement.

Railroad police are to be certified in their home state and have nationwide jurisdiction by federal law, on the property and rights-of-way of their employer as well as in connection with its services (conceivably quite a wide scope as railroads run right through virtually all major cities--the CPD "bait truck" incident that cause controversy a few years ago involved NSRR Police). With the exception of Amtrak, these are all private companies.

I'm not sure to what extend qualified immunity does or does not apply to them, however.

PA definitely has something like this too.. just like in NC, it originated from company towns, in their case mining.

Yes. They're actually police, with attendant legal status and powers.