In order to answer this question we have to agree on the meaning of point-like. (This is not so obvious since nature happens to be quantum rather than classical)
In practice, one has to specify a framework where the definition can be operationally, at least in principle, tested against the experimental evidence.

The tentative definition that I will adopt in the following is: a particle is point-like if every physical process (say a scattering), at any energy scale (or kinematical configuration) above a certain threshold, agrees with the prediction made by a perturbative renormalizable quantum field theory where the particle is elementary.
An equivalent definition could be that the action for such a particle is dominated by its free kinetic energy at all scales above a certain threshold (or, again, that the theory is always around a Gaussian fixed point). In practice I am trading point-like for elementary which is a (slightly) better defined concept.

I had to include the notion of perturbativity to speak of particles in the first place, that is of (presumably effective) field theories that are close to a gaussian fixed point in at least a finite energy range.
This definition is not perfect, but it makes clear that a theory of particles strongly interacting at all scales isn't in fact a theory of particles after all.

The proton isn't elementary because its interactions at or above the confinement scale are strongly coupled and, moreover, the theory would require infinitely many terms in the lagrangian making it non-renormalizable too. The pions, on the other hand would seem to be elementary at small energy (essentially because of their Goldstone boson nature and Adler's theorem) but the interactions become strong again at $E\sim\Lambda_\textrm{QCD}$. The interactions are non-renormalizable too. In fact, the requirement of non-renormalizability and the strong interactions usually go together in concrete realizations of compositeness.

Buying this tentative definition for point-like, we can ask whether the electron is so. The answer is yes: it is point-like, to the best of our present knowledge. In other words, up to the energy scale of the order of few tens of $\mathrm{TeV}$'s that we have been able to explore experimentally (the precise number depends on various things that would take us very far), there is no sign that the electron isn't described by the renormalizable weakly coupled quantum field theory known as the Standard Model at all scales above the $\Lambda_\mathrm{QCD}$. In such a theory the electron is an elementary field.

Various caveats are in order. First, I am neglecting gravity which makes the SM non-renormalizable (and gravity may becomes strong at $M_\textrm{Planck}$). In the leading quantum theory of gravity that explains the dynamics at the Planck scale, string theory, the electron isn't quite a particle nor point-like. The Planck lenght is however so small that we can safely ignore this point for most of the questions. Second, the gauge coupling for the hypercharge in the Standard Model is believed to have a Landau pole that may break the theory at even larger energy scales than Planck. Hence, one can safely neglect the Landau pole too (quantum gravity effects kick-in much earlier).

Say one day we discover a discrepancy between the predictions of the Standard Model (SM) concerning the electrons and the experimental data. To be concrete, imagine one day we discover a 5~$\sigma$ discrepancy in the $g_{e}-2$ of the electron. Would that mean that the electron is composite? No, at least non-necessarily. In fact, the extra corrections $\delta_\textrm{BSM}$ in $(g_{e}-2)=\delta_\textrm{SM}+\delta_\textrm{BSM}$ could be accounted for a new weakly coupled renormalizable field theory valid above a new threshold (the mass of the new particles involved in producing $\delta_\textrm{BSM}$) where the electron is still an elementary field. There exist several models beyond the SM where this is the case: they go beyond the SM coupling new weakly interacting particles to the electron, changing some of its low energy properties; however, above the mass of these new states the electron is still accounted as an elementary particle coupled weakly to the old fields and few new ones. On the other hand, the $\delta_{BSM}$ could be explained by the electron being compositeness, i.e. non point-like. This would be the correct explanation should the new weakly coupled renormalizable theory expressed in terms of other fields than the electron. One could still insist to use the electron above the compositeness scale but the theory would be strongly interacting and non-renormalizable, in such a variable.

This post imported from StackExchange Physics at 2017-11-23 19:18 (UTC), posted by SE-user TwoBs