I am stuck in constructing a function that is locally Lipschitz continuous at $x_0$ but it does not have directional differentiation at $x_0 $ in any direction.
The definition of directional derivative is $$f'(x_0,h)=\lim_{t\downarrow 0}\frac{f(x_0+th)-f(x_0)}{t}$$
Depending on the interpretation of directional derivative, $F(x)=|x|$ could be such an example with $x_0=0$. It has no two-sided derivative along any line. But I suppose you meant one-sided directional derivatives.
Begin by constructing a Lipschitz function $f:[0,1]\to\mathbb R$ such that $f'(0)$ does not exist. For example, $f$ can be a piecewise linear function with slope $\pm 2$ which changes slope at the points where $|f(x)|=|x|$. In other words, the graph of $f$ is a zigzag of slope $\pm 2$ that hits both lines $y=x$ and $y=-x$ infinitely many times.
Or, if you prefer a more formulaic approach, let $f(x)=x\sin\log x$. Since $f'(x)=\sin\log x+\cos\log x$ is bounded, $f$ is Lipschitz.
Then let $F(x)=f(|x|)$.