Azumaya originally defined an Azumaya algebra (which he called a proper maximally central algebra) to be an algebra A which is a free module of finite rank over its centre Z such that the natural map $$A\otimes_Z A^{\mathrm{op}}\to \mathrm{End}_Z(A)$$ is an isomorphism. More modern definitions (e.g., Knus, Quadratic and Hermitian Forms over Rings) replace "free" with "faithfully projective". Is there an example of an algebra which meets this broader definition but does not meet the original one? I.e., an Azumaya algebra that is faithfully projective but not free as a module over its centre. If possible, I'd also like the centre to be a noetherian $k$-algebra, for some algebraically closed field $k$.

Even better would be some method for producing lots of examples of Azumaya algebras "to order", but that's probably asking too much!