A 3D Packing Question

Basic Question: Is there a convex polygon that can tile the plane only such that the arrangement is not vertex to vertex? Note: In a vertex to vertex arrangement, any vertex of any unit touches any other unit only at a vertex and not at an intermediate point of an edge.

The answer appears to be yes. See this: https://en.wikipedia.org/wiki/Marjorie_Rice
3 of the 4 pentagonal tiles discovered by Marjorie Rice appear to tile only in a non-vertex-to-vertex fashion. However, the three non-regular hexagons that tile the plane (discovered by Reinhardt; see this: https://arxiv.org/pdf/1803.06610.pdf ) appear to give v-to-v tiles.
And of course, all triangles and all quadrilaterals tile but they all can tile the plane in v-to-v manner.

Question: Going on to 3D, are there polyhedral units (convex or otherwise) that can pack 3D space without gaps only such that the arrangement is NOT vertex to vertex or edge to edge?