Can you say a piece of soap? (I don't speak english fluently)
Well, what you would buy in a store is "a bar of soap" (kostka mydła), but when you use some of it and perhaps even break a part of it, you could say that it's a piece of soap.
Czy "kawałek" może też oznaczać cząstkę układanki?
Tak. Ogólnie tak. Choć jeśli mamy do czynienia z puzlami to pojedynczy element nazywamy puzlem.
Just don't drop it!
Why isn't "bar" accepted? I can't see what would be a bit of soap but not qualify as a bar of soap.
"kawałek" isn't idiomatic. It must be literally "a piece" of soap. "a bar of soap" that you buy in a shop is "kostka mydła".