  We start a systematic study of non-projected supermanifolds, concentrating on
supermanifolds with fermionic dimension 2 and with the reduced manifold a
complex projective space. We show that all the non-projected supermanifolds of
dimension $2|2$ over $\mathbb{P}^2$ are completely characterised by a non-zero
1-form $\omega$ and by a locally free sheaf $\mathcal{F}$ of rank $0|2$,
satisfying $Sym^2 \mathcal{F} \cong K_{\mathbb{P}^2}$. Denoting such
supermanifolds with $\mathbb{P}^{2}_\omega(\mathcal{F})$, we show that all of
them are Calabi-Yau supermanifolds and, when $\omega \neq 0$, they are
non-projective, that is they cannot be embedded into any projective superspace
$\mathbb{P}^{n|m}$. Instead, we show that every non-projected supermanifolds
over $\mathbb{P}^2$ admits an embedding into a super Grassmannian. By contrast,
we give an example of a supermanifold $\mathbb P^{2}_\omega(\mathcal F)$ that
cannot be embedded in any of the $\Pi$-projective superspaces $\mathbb
P^{n}_{\Pi}$ introduced by Manin and Deligne. However, we also show that when
$\mathcal F$ is the cotangent bundle over $\mathbb{P}^2$, then the
non-projected $\mathbb{P}^2_\omega(\mathcal F)$ and the $\Pi$-projective plane
$\mathbb P^{2}_{\Pi}$ do coincide.
