If $0\notin\partial f(x_k)$ then any $-\xi\in-\partial f(x_k)$ is a descent direction.
Solution 1:
Let $f(x) = \max(100x_1+x_2,-100x_1+x_2)$. Then $\partial f(0) = \operatorname{co} \{ (100,1), (-100,1) \}$, but $d=(-100,-1)$ is not a descent direction at $x=0$ since $0+10000t-t > 0$ for all $t>0$ and so $f(0+td)>0$ for all $t>0$.
Note that the nearest point to $\partial f(0)$ is $(0,1)$ and $d=-(0,1)$ is a descent direction.