Is there a finite object whose charge distribution is a perfect dipole (only has a dipole moment) yet doesn't involve any infinities or discontinuities in the fields? And if there happens to be a fairly "standard" object in textbooks or literature, that would be even better.
One example of a perfect dipole is a sphere of material with a uniform polarization. Outside of the sphere, in a multipole expansion the potential only has a dipole moment, so this configuration looks like a perfect dipole externally.
Another example of a perfect dipole is a pair of opposite charges ($\pm q$) in the limit that the distance between them ($d$) goes to zero but their charge increases such that $p = qd$ remains constant.
Neither of these objects is easy to analyze rigorously. The uniformly polarized sphere at least doesn't involve a limit procedure with charges going to infinity, but it does have discontinuities in the electric field at the surface (although the potential is still continuous; I was initially incorrect on that).
In case any such object is very complicated, and so the motivation behind the question would help frame an answer, I was thinking about this after reading Electromagnetic field energy "paradox" and wondering if the answer is hidden in the discontinuities of the field.