4. The directrix of the parabola $y=-\frac{1}{8} x^{2}$ intersects the $y$-axis at point $A$. A line through $A$ intersects the parabola at points $M$ and $N$. Point $B$ is on the axis of symmetry of the parabola, and $\left(\overrightarrow{B M}+\frac{\overrightarrow{M N}}{2}\right) \perp \overrightarrow{M N}$. Then the range of $|\overrightarrow{O B}|$ is ( ).
(A) $(3,+\infty)$
(B) $(4,+\infty)$
(C) $(5,+\infty)$
(D) $(6,+\infty)$
4.D.

Notice the point $A(0,2)$. Draw a line $M N$ through $A$, with its equation set as $y=k x+2$. Substituting into the parabola equation yields
$$
x^{2}+8 k x+16=0 \text {. }
$$

And $\Delta>0$, then $k^{2}>1, x_{1}+x_{2}=-8 k$.
Let point $B(0, b)$, and the midpoint of $M N$ be
$$
\begin{array}{l}
C\left(-4 k,-4 k^{2}+2\right) . \\
\text { By }\left(\overrightarrow{B M}+\frac{\overrightarrow{M N}}{2}\right) \perp \overrightarrow{M N} \\
\Leftrightarrow B C \perp M N \Rightarrow b=-4 k^{2}-2<-6 .
\end{array}
$$

Then the range of $|\overrightarrow{O B}|$ is $(6,+\infty)$.