[Lamadrid 63-1] shows the counterexample of the Banach property of the injective tensor norm.
https://www.ams.org/journals/bull/1963-69-06/S0002-9904-1963-11037-X/S0002-9904-1963-11037-X.pdf
I wonder there is some kind of "minimal tensor products" of Banach algebra and generalized nuclearity like $C^{*}$-algebras, operator spaces or locally convex spaces; that is the equivalence of having unique tensor products and having nice approximation property.