Let $V$ be an infinite dimensional topological vector space and consider the natural application $\iota\colon V\to V^{**}$. The space $V$ is said to be reflexive if $\iota$ is an isomorphism.

Are there examples where $\iota$ fails to be an isomorphism but $V$ and $V^{**}$ are nevertheless isomorphic?

Can one find an example where $V$ is a Banach space and the isomorphism is actually an isometry?