Is there an example of a complex analytic space $X$ that doesn't have any (not necessarily open or closed) positive dimensional subspace $Y$ which is analytically isomorphic to (the complex analytic space associated to) a scheme?

**Edit:** after D.Arapura's comment, we require $X$ to have dimension $>1$.

If I remember correctly, there are non "Abelian" complex tori $X=\mathbb{C}^n/\mathrm{Lattice}\;\;$ that do not have any positive dimensional analytic subvariety. Can a counterexample be derived from this?

Also, any complex algebraic space has an open subspace which is a scheme. So the counterexample (if it exists) must be searched outside algebraic spaces.

What if the question is modified by requiring that $X$ has no $Y$ that is locally closed in the

analyticZariski topology (where opens are complements of analytic subspaces)?