Are there any small banks or businesses that might finance this, even at a higher than normal interest rate?
Many, but some of them are ones you wouldn't want to deal with. One reputable one is Oppenheim Kredit on Gornoslaska in Warsaw who used to be very good about sourcing credit for people without Polish citizenship who want to buy real estate. The amount you want might be a bit small and I'm not sure how the credit crunch has affected them, however they might be worth checking out.
Why not raise the finance in your home country? Or just rent in Poland?