Does divergence of Taylor series for relativistic $E(p)$ for $p\ge m$ have any physical significance?

When one wants to include weak relativistic effects in classical equations, usually kinetic energy term is expanded into Taylor series about $p=0$. But the complete dispersion relation is $E=\sqrt{m^2+p^2}$, which has branch points at $p=\pm im$, thus its Taylor series diverges for any $p: |p|\ge m$.

Does such divergence have any physical significance? Does it mean that there're some completely new physical effects starting right at $p\approx m$, and trying to just include more complete dispersion relation will lead to nonsensical results? Maybe it is somehow related to possibility of pair production/annihilation?

Physically speaking, this means that the approximation for low momentum (I have added the speed of light, $c$),
$$E=mc^2+\frac{p^2}{2m},$$
is not valid at high momentum. I would not attribute too much physical meaning to the value of $|p|$ for which it starts diverging.