Do there exist positive integers $a,b$ and a prime $p>\max(a,b)$ such that $p^3$ divides $(a+b)^p-a^p-b^p$?

The reader of Kvant magazine A. T. Kurgansky asked to prove that such $a,b,p$ do not exist, see here. But discussion here

On the exact reference of a cute Diophantine problem

suggests that it should be very hard to prove. Maybe, a counterexample may be bound? Roughly speaking, a probability of this event is about $1/p^2$, for each $p$ we have about $p$ events (even for $b=1$, I was previously wrong that it may be supposed without loss of generality, thanks for Noam Elkies for noting this) and so as $\sum 1/p=\infty$, we may expect them (and even infinitely many!) But this series converges very slowly, so the minimal example may be large.