Let $X$ be a Banach space.
By Banach-Alaoglu and Krein-Milman Theorems, one can show that if $X$ is a dual space, then $X$ must have at least one extreme point of the closed unit ball.

I am interested in its converse.
More precisely, 
> Question: Let $X$ be a Banach space. 
If the closed unit ball of $X$ has at least one extreme point, must $X$ be a dual space?

I feel that the statement above is negative. 
However, I could not produce a counterexample.

In fact, the only Banach spaces which I know that  are not dual spaces are $c_0$ and $C_0(\mathbb{R})$ (the latter set is the collection of all real-valued continuous function vanishing at infinity) because both sets have no extreme point.