Is dihydrogen the only example of overlap between two s orbitals?
You don't need particularly exotic things to get a counterexample. Metal–metal bonding in mercury(I) compounds is quite prevalent, where the electronic configuration is $\ce{Hg+}$: $\ce{[Xe](4f^14)(5d^10)(6s^1)}$, and the bonding involves 6s–6s overlap. The prototypical example is the $\ce{Hg2^2+}$ cation, which features in mercury(I) chloride $\ce{(Hg2^2+)(Cl^-)2}$; this cation is valence isoelectronic to $\ce{H2}$.

Fundamentally, it is a relativistic contraction of the 6s orbital which makes such bonding common in Hg(I): see also Why does mercury form polycations?. The bonding in alkali metal dimers is qualitatively similar, but significantly weaker.

If we play the "every bond has orbital overlap" card (i.e. not just covalent bonds, which I think is perfectly fair), then the delocalised bonding in Group 1 and 2 metals arises from overlap of ns orbitals on each atom to form bands, as has already been alluded to in the comments. Even simpler would be lithium hydride, which likely has some degree of 1s–2s orbital overlap, even though the bonding may be primarily ionic.