The title is pretty much the complete question. An example of this would (I believe) have to be a nonconvex polycube. If there is one, I am also interested in the smallest example.
The analogous question for other planar polyforms, such as polyabolos, is also of interest.
The smallest example I have found so far contains $16$ cubes:
It has six rectangular faces of dimensions $1\times 1, 1\times2, 1\times3, 1\times4, 2\times3,$ and $3\times3,$ as well as four nonconvex faces.
After writing some code, I've confirmed that every polycube with at most $10$ cells lacks this property, so $11$ is a lower bound.