The random k-SAT model is the most important and well-studied distribution over k-SAT instances. It is closely connected to statistical physics and is a benchmark for satisfiability algorithms. In this paper, we prove that any Cutting Planes refutation for random k-SAT requires exponential size, for k that is logarithmic in the number of variables, and in the interesting regime where the number of clauses guarantees that the formula is unsatisfiable with high probability.