No. Let $X$ and $Y$ be Banach spaces, and set $Z=X\oplus Y$, with $\||(x,y)|\|:=\|x\|+\|y\|$. Assume that $x$ is a extreme point of $X$ with $\|x\|=1$. Then $(x,0)$ becomes an extreme point of $Z$; indeed, if $$(x,0)=\frac12(a,y)+\frac12(b,z)$$ for $(a,y),(b,z)$ in the unit ball of $Z$, we then have $a=x=b$, since $x$ is an extreme point, but then $1=\|x\|=\|a\|\leq\||(a,y)|\|\leq 1$, so $y=0$, and analogously, $z=0$.
So, $L^2(\mathbb R)\oplus L^1(\mathbb R)$, is not a dual space, but its unit ball has extreme points.