Completely entangled subspaces of entanglement depth $k$

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.