Quiz

Sure. It will be a concave polyhedron that will rotate from faces to edges.

It needs to be possible to place it on "any face," but a concave shape generally cannot be placed on certain faces.

Added 42 minutes later
If the polyhedron is placed on an edge (or on a vertex), such a balance will not be stable (because the center of gravity will be above the support), and the polyhedron will fall onto one of the faces. If, at the same time, the projection of the center of gravity turns out to be within the area of the table (or wherever the polyhedron is placed) on which this face lies, then the polyhedron, placed on this face, will not fall. Such a face will always be found (the polyhedron will "fall" until it "finds" such a face), so the answer is no.

P.S. Of course, if the polyhedron is convex, but it seems that the question implies that it is convex. Otherwise, it is generally impossible to place it on "any face."

Added 12 minutes later

if the plane is really flat, then gravity + friction - and the ball will not roll anywhere.

But the ball will roll very well. Except in cases where its center of gravity lies on a diameter perpendicular to the top of the table.