Here is an example:

In X13 of "Faisceaux amples sur les schemas en groupes", Raynaud provides an example of a group scheme *G -> S* in chacteristic 2 where

*S* is a local regular scheme of dimension 2.
*G -> S* is smooth, separated and quasi-compact.
- The fibers of
*G ->S* are affine and the generic fiber is connected.

such that *G -> S* is not quasi-projective.

Therefore, the classifying stack BG has affine stabilizers but does not have a quasi-affine diagonal.

On the other hand, in VII 2.2, Raynaud proves that if *G -> S* is a smooth, finitely presented group scheme such that

*S* is normal.
*G -> S* has connected fibers.
- The maximal fibers are affine.

then *G -> S* is quasi-affine.

Question: Is the above statement true if (2) is weakened to require that the number of connected components over a fiber *s \in S* be prime the characteristic of the residue field *k(s)*?

Of course, one would really like to know if the statement is true if *G->S* is not necessarily flat so that one could apply it to the inertia stack.

On a related note, Raynaud also provides an example in VII3 of a smooth quasi-affine group scheme *G ->* **A^2** over a field k with connected fibers but which is not affine. The classifying stack BG gives an example of stack with affine and connected stabilizers but with non-affine inertia stack. In the example of a scheme with non-affine diagonal, the inertia is of course affine. It's also easy to provide examples of non-affine group schemes with affine but non-connected fibers (eg. the group scheme obtained by removing the non-identity element over the origin from **Z/2Z** *->* **A^2**).