11. In the right triangular prism $A_{1} B_{1} C_{1}-A B C$, it is known that $\angle B A C=\frac{\pi}{2}, A B=A C=A A_{1}=1, G$ and $E$ are the midpoints of $A_{1} B_{1}$ and $C C_{1}$, respectively, and $D$ and $F$ are moving points on segments $A C$ and $A B$ (excluding the endpoints). If $G D \perp E F$, then the range of the length of segment $D F$ is $(\quad)$.
(A) $\left[\frac{1}{\sqrt{5}}, 1\right)$
(B) $\left[\frac{1}{5}, 2\right)$
(C) $[1, \sqrt{2})$
(D) $\left[\frac{1}{\sqrt{5}}, \sqrt{2}\right)$
11. A.

Establish a rectangular coordinate system, then
$$
\begin{array}{l}
F(m, 0,0)(0<m<1), D(0, n, 0)(0<n<1), \\
G\left(\frac{1}{2}, 0,1\right), E\left(0,1, \frac{1}{2}\right) . \\
\text { Hence } \overrightarrow{G D}=\left(-\frac{1}{2}, n,-1\right), \overrightarrow{E F}=\left(m,-1,-\frac{1}{2}\right) .
\end{array}
$$

From the problem, we know
$$
\begin{array}{l}
\overrightarrow{G D} \cdot \overrightarrow{E F}=-\frac{m}{2}-n+\frac{1}{2}=0 \\
\Rightarrow m+2 n=1 \\
\Rightarrow 0<n<\frac{1}{2} . \\
\text { Also, } \overrightarrow{D F}=(m,-n, 0) \text {, then } \\
|\overrightarrow{D F}|^{2}=m^{2}+n^{2}=(1-2 n)^{2}+n^{2} \\
=5 n^{2}-4 n+1 .
\end{array}
$$

When $n=\frac{2}{5}$, $D F$ reaches its minimum value $\sqrt{\frac{1}{5}}$; when $n=0$, $D F$ reaches its maximum value 1.