  We introduce a class of entangled subspaces: completely entangled subspaces
of entanglement depth $k$ ($k$-CESs). These are subspaces of multipartite
Hilbert spaces containing only pure states with an entanglement depth of at
least $k$. We present an efficient construction of $k$-CESs of any achievable
dimensionality in any multipartite scenario. Further, we discuss the relation
between these subspaces and unextendible product bases (UPBs). In particular,
we establish that there is a non-trivial bound on the cardinality of a UPB
whose orthocomplement is a $k$-CES. Further, we discuss the existence of such
UPBs for qubit systems.
