Is there a sequence $(x_n) \subset E$ such that $|x_n|=1$ for all $n$ and that $x_n \to 0$ in $\sigma(E, E^*)$?

Question

Let $(E, |\cdot|)$ be an infinite-dimensional Banach space and $\sigma(E, E^*)$ its weak topology.

Is there a sequence $(x_n) \subset E$ such that $|x_n|=1$ for all $n$ and that $x_n \to 0$ in $\sigma(E, E^*)$?

In case $E^*$ is separable or $E$ is reflexive, the answer is affirmative. If the answer is negative, we fix $\varepsilon >0$.

Is there a sequence $(x_n) \subset E$ such that $|x_n| \ge \varepsilon$ for all $n$ and that $x_n \to 0$ in $\sigma(E, E^*)$?

Thank you so much for your elaboration!