  Genuinely entangled subspaces are a class of subspaces in the multipartite
Hilbert spaces that are composed of only genuinely entangled states. They are
thus an interesting object of study in the context of multipartite
entanglement. Here we provide a construction of multipartite subspaces that are
not only genuinely entangled but also fully non-positive-partial-transpose
(NPT) in the sense that any mixed state supported on them has non-positive
partial transpose across any bipartition. Our construction originates from the
stabilizer formalism known for its use in quantum error correction. To this
end, we first introduce a couple of criteria allowing to assess whether any
state from a given non-trivial stabilizer subspace is genuinely multipartite
entangled. We then use these criteria to construct genuinely entangled
stabilizer subspaces for any number of parties and arbitrary local dimension
and conjecture them to be of maximal dimension achievable within the stabilizer
formalism. At the same time, we prove that every genuinely entangled subspace
is fully NPT in the above sense, which implies a quite surprising fact that no
genuinely entangled stabilizer subspace can support PPT entangled states.
