  In a multipartite scenario quantum entanglement manifests its most dramatic
form when the state is genuinely entangled. Such a state is more beneficial for
information theoretic applications if it contains distillable entanglement in
every bipartition. It is, therefore, of significant operational interest to
identify subspaces of multipartite quantum systems that contain such properties
apriori. In this letter, we introduce the notion of unextendible biseparable
bases (UBB) that provides an adequate method to construct genuinely entangled
subspaces (GES). We provide an explicit construction of two types of UBBs --
party symmetric and party asymmetric -- for every $3$-{\it qudit} quantum
system, with local dimension d\ge 3. Further, we show that the GES resulting
from the symmetric construction is indeed a {\it bidistillable} subspace, i.e.,
all the states supported on it contain distillable entanglement across every
bipartition.
