Could you give me an example of a compact Kähler manifold which analytically deforms to a non Kähler one?

For example, there is no hope to find a complex structure on a Hopf manifold in order to make it Kähler because of topological obstructions (the second Betti number is zero).

For instance, I think that the Iwasawa manifold should not have topological obstructions.

Of course, the algebraic counterpart of my question has an affirmative answer: complex tori of dimension greater than one give examples of manifolds that can be analytically deformed from an algebraic one to a non algebraic one (but still Kähler).

Thanks in advance!