I've seen Tychonoff's theorem be used to prove that the $ p $-adic integers are compact. The proof is easy: there is a natural embedding

$$ \mathbb Z_p \to \prod_{k=1}^{\infty} (\mathbb Z/p^k \mathbb Z) $$

whose image is closed, and the infinite product is compact by Tychonoff, so in particular we deduce that $ \mathbb Z_p $ is compact. (This strategy is used in general to show other profinite objects are compact, for instance, infinite Galois groups under the Krull topology.)

The use of Tychonoff (and by extension the axiom of choice) is unnecessary: we can simply adapt the usual proof of Heine-Borel over $ \mathbb R $ to show that $ \mathbb Z_p $ is compact. If there is an infinite open cover with no finite subcover, we can find an infinite descending chain of closed balls in $ \mathbb Z_p $ intersecting at a single point that need infinitely many open balls to cover them, and since an open ball including the single point will cover all sufficiently small closed balls including that point, we get a contradiction.