Denote $a=VA,b=VB,c=VC$, then $3p=a+b+c+\sqrt{a^{2}+b^{2}}+\sqrt{b^{2}+c^{2}}+\sqrt{c^{2}+a^{2}}$. Now, use AM-GM (note: volume of tetrahedron equal $\frac{abc}{6}$).

The same problem appeared already in USAMO 1976. As N.T.Tuan did not give the result, the maximal volume is $ \dfrac{p^3}{6\cdot(\sqrt {2} + 1)^2}$ with inequality if and only if $ a = b = c$.