After going through several references, I found out that the Lefschetz isomorphism that you mentioned above seems to work only for the projective case. Would you care to tell a reference (if there is) where this has been proven in the proper case?

Take a look at M. Hazewinkel's Formal Groups and Applications. I think you'll find a proof or (at least) an idea of how you can prove the theorem.

Thank you very much! This is quite the answer I was looking for. I was actually more interested in the the case when $i$ is odd. A colleague told me that when $i$ is odd, $K(V) \supset K(\mu_{p^\infty})$ only after a finite extension of $K$; but he can't remember the proof nor a reference that gives such a proof.

...I thought it was better to know first what it means to have good ``supersingular" reduction before asking questions about the action of Galois on $V$.

Thanks Felipe and Matt for your comments. As for Felipe's question, I am really interested in the Galois representation $V = H^i_{et}(X_{\bar{K}}, \mathbb{Q}_p)$. When $X$ has good ordinary reduction, we know how the inertia subgroup of the absolute Galois group $G_K$ acts on $V$ (cf. article of B. Perrin-Riou and its appendix by Illusie in Asterisque 223). I was wondering whether whether there is a known description about the action of $G_K$ on $V$ when $X$ has good ``supersingular" reduction. But...

Apologies for the confusion. Although I am more interested in a more general case, I will now consider $K$ to be a finite extension of $\mathbb{Q}_p$. This will also enable me to understand what happens in a basic case.

Ooops. Sorry about the first line. What I meant was that $K$ is a field of characteristic not equal to $p$. Thanks for your answer. So, upon reading your answer, it seems to me that it doesn't matter weather $u$ is a unit that is not a root of unity or not. The extension will be non-abelian as long as it is not a $p$th power. Moreover, what about when $K$ contains some $p^n$th root of unity?