More elementary than Ascoli:

If it was normable, it would mean that there exists a norm $n$ which is continuous, hence for some $k$, $n(\phi) \leqslant C\ p_k(\phi)$ and which defines the topology, i.e. such that for all $k$, $p_k(\phi) \leqslant C_k\ n(\phi)$.

This would imply that all the norms $p_k$ are equivalent for $k$ large enough, which is not the case: for example $\exp(-|x|^2) \sin(N x_1) /N^k$ tends to $0$ for the first $k$ norms but not for the following.