Whisk smoked paprika, brown sugar, garlic powder, salt and black pepper together in a large glass or ceramic bowl. Add pork and toss to evenly coat. Cover the bowl with cling film and marinate in the fridge for 30 minutes.
Turn on a multi-functional pressure cooker (such as Instant Pot®) and select Saute function. Heat oil and cook pork in batches until browned, 5 to 7 minutes. Pour in barbecue sauce and water. Close and lock the lid; select Manual function and set timer for 60 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure using the natural-release method according to manufacturer's instructions, 10 to 40 minutes. Unlock and remove lid; transfer pork to a separate plate. Reselect Saute function and simmer sauce until thickened, about 10 minutes. Shred pork with 2 two forks; mix with sauce.