I would argue that looking for a ring spectrum is not the right thing to do. What you should be looking for is the category of modules over that possibly non-existent ring spectrum, as an infinity-category or just as a triangulated category. If you think about it this way, an obvious answer presents itself.
Begin with the category of HZ-modules, or the derived category of Z, or its infinity-category version. Now take the Bousfield localization with respect to the object Fp (thought of as a complex in degree 0). This is not a smashing localization, so this category is not equivalent to modules over HZp . As a triangulated category, it is compactly generated, but by Fp itself. The sphere is not small. So this category is equivalent to modules over a DGA, the endomorphism DGA of Fp, but it is not commutative. It is more like a DG Hopf algebra, I suspect, so that its homotopy category has a tensor product even though it is not commutative. I have always thought this example needs more investigation, though it might be in Dwyer-Greenlees-Iyengar somewhere. It is a toy version of the K(n)-local stable homotopy category.