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Let $p$ be a prime and $K$ a $p$-adic field (a finite extension of the field of $p$-adic numbers ${\mathbb{Q}}_p$). We employ the main results in [12] and the arithmetic of elliptic curves over $K$ to reduce the problem of classifying 3-dimensional non-associative division algebras (up to isotopy) over $K$ to the classification of ternary cubic forms $H$ over $K$ (up to equivalence) with no non-trivial zeros over $K$. We give an explicit solution to the latter problem, which we...