Let $p$ be any prime number and $K$ any finite extension of $\mathbb{Q}_p$. The torsion part of $\mathcal{O}_K^*$ is always cyclic. On the other side, the torsion part of $\mathbb{Z}_p^{*} \oplus \mathbb{Z}_p^{*}$ is isomorphic to $$\mathbb{Z}/(p-1)\mathbb{Z} \oplus \mathbb{Z}/(p-1)\mathbb{Z}$$ which is not cyclic. See Henri Cohen's Number Theory, volume 1, section 4.3. There, he gives an explicit descritption of the group of units of $\mathcal{O}_K$. If I remeber correctly, this also appears in Iwasawa's book on Local Class Field Theory.

Let $p\ge3$ be a prime number and $K$ any finite extension of $\mathbb{Q}_p$. The torsion part of $\mathcal{O}_K^*$ is always cyclic. On the other side, the torsion part of $\mathbb{Z}_p^{*} \oplus \mathbb{Z}_p^{*}$ is isomorphic to $$\mathbb{Z}/(p-1)\mathbb{Z} \oplus \mathbb{Z}/(p-1)\mathbb{Z}$$ which is not cyclic. See Henri Cohen's Number Theory, volume 1, section 4.3. There, he gives an explicit descritption of the group of units of $\mathcal{O}_K$. If I remeber correctly, this also appears in Iwasawa's book on Local Class Field Theory.

Edit: I forgot the condition $p\ge3$, thanks Keith.