A related question was asked earlier by Mark Meckes: [Self-dual finite-dimensional complex normed spaces](https://mathoverflow.net/questions/123124/self-dual-finite-dimensional-complex-normed-spaces). He pointed out the $l^1$-$l^\infty$ example and noted that it generalizes to unit balls that are regular polygons in the real two-dimensional case. He also told us the $X \oplus_2 X^*$ construction.

I believe the specific question asked by Wlodzimierz has a positive answer, based on a comment I heard Giles Pisier make many years ago --- he said something very similar to this, though I don't remember exactly what. I don't have a reference though.