MOST likely the Piston Ring & Cylinder will have to be replaced as this is by far the most common part replaced on these oil-less units. If the teflon-coated Piston Ring has a small tear, fold or scratch it will cause air to pass through and it won't compress to the maximum PSI where the unit can shut off.

I would take the pump head off first and inspect this. It could be as simple as one of your gaskets is damaged/torn or a valve is cracked, but that's obvious once you take the head off as you'll see the flat, metal reed valve and will be able to tell if it's rusted badly or cracked.

Unfortunately, no piston parts are available for this pump any longer.

As long as you don't have any other major air leaks externally, this is probably the culprit.
Hope this helps!
- MTR