I was looking carefully at all the definitions, trying to understand exactly what was going on in this question on categorical duals in Banach spaces. It seems that in the category of Banach spaces with contractions, only finite dimensional Banach spaces have duals in the sense of category theory. This is because one needs morphisms
$$ I \to A \otimes A^* $$
$$ A \otimes A^* \to I $$
where the first is like the inclusion of the identity and the second like the trace. This means that the identity map on $A$ has to have a trace, and that only works for finite dimensional Banach spaces.
So, my question: am I right? Or have I misunderstood something here?