Drilled into white/cream colored plastic behind drywall, what is it?
No, it's not safe to drill into something unidentified. 

White PVC is most commonly used for waste water (and vents, which are connected to the waste water lines but carry sewer gas to a vent the roof) and central vac. Schedule 20 is used for central vac and is much thinner than the Schedule 40 used for plumbing



It's also possible it's a conduit run for future use, eg running wires from the basement to the attic -- which may or may not have something in it now. 

Now that you've hit it, you really need to open up the drywall there and figure out what you're dealing with, repair it, and then patch up the drywall. I'd suggest just opening a relatively small hole first to inspect it, then cut a bigger one if needed once you know what you need to do. 

If it's central vac or plumbing, you'll have cut out the damaged section and replace it. If it's small hole, you may just be able to cut just that tiny bit out and replace it with a single coupling. For a bigger hole you'll have to replace with two couplings.

You can probably post back with a picture of the damage and describe what you see inside and we can provide better insight into what it is and how to fix it.