It would most likely be impossible to come up with a reliable measure of the volume of an orbital.
Orbitals are defined over all space, so an integral like this
\iiint\phi(x,y,z)\,\mathrm dx\,\mathrm dy\,\mathrm dz
is not guaranteed to be finite, and also doesn't really mean anything. Clearly, the orbitals are decaying, but are they decaying quickly enough to have this integral be finite? I would suspect they probably are because the hydrogen atom orbitals decay like $\exp(-r)$, and this integral from $0\rightarrow\infty$ is finite. Regardless, $\phi$, is an atomic orbital here, but the integral over the space of an atomic orbital, molecular orbital, or wavefunction doesn't mean anything physically that I am aware of. Only the integral over the magnitude squared.
One could think about picking a fixed distance from the nucleus, and then integrating over all space inside this distance from the nucleus, but this would always be arbitrary. I doubt that a $p$ orbital would ever be one third the volume of an $s$ orbital for a reasonable choice of distance.
Third, any measure of the volume of an orbital which gives a finite answer is going to be spoiled when you consider that the shape or orbitals is not unique in the sense that a unitary transformation of the molecular orbitals gives another set of valid molecular orbitals. These transformations will almost certainly not maintain the same volume within some distance of the nucleus or anything like this.
It is perhaps possible to quantify what this lecturer is getting at in a more concrete way, however. For instance, there are different schemes of creating localized molecular orbitals in the context of Hartree-Fock. The Edmiston-Reudenberg method of orbital localization maximizes the electron-electron repulsion. So, one could carry out this procedure to generate orbitals which look like the typical $s$ and $p$ orbitals we think about, then look at the result of this maximization procedure to verify that the electron-electron repulsion is larger in the $p$-like orbitals than the $s$-like orbitals. This might work, but because the shape of the orbitals can easily be changed, I doubt this measure has any chemical significance.
So to summarize, the lecturer's reasoning may not be strictly wrong, but she is being very loose with objects that are poorly defined, such as the volume of an orbital. I also explained why the volume of an orbital is poorly defined, and there is really no way to fix this completely.
As to the question she was actually trying to answer, inter-electronic repulsion in $p$ orbitals versus $s$ orbitals is completely unnecessary. Here is a graph of ionization energy with atomic number
It is quite clear that the dips in the graph correspond to time when we go from a half-filled shell to one extra electron (nitrogen to oxygen) or from a full shell to the first electron in an empty shell (beryllium to boron). The beryllium to boron case says this cannot possibly be due to inter-electronic repulsion being stronger in $p$ orbitals because there is only one electron in the $p$ orbital of boron. I would say most of the explanation of this phenomenon is answered in this question. Basically, it boils down to the fact that these electrons which are added to a new shell or half-full shell must be the opposite spin of the other electrons, so they are not stabilized by exchange, and are higher in energy despite the greater attraction to a more positively charged nucleus. Notice this effect decreases as you go down the periodic table. This is understood by noting that exchange is a somewhat distance-dependent phenomenon, so as the orbitals become larger, this loss of stabilization from losing exchange is not as competitive with the increased nuclear charge.