  We define Sobolev spaces $H^{\mathfrak{s}}(K_q)$ over a local field $K_q$ of
finite characteristic $p>0$, where $q=p^c$ for a prime $p$ and $c\in
\mathbb{N}$. This paper introduces novel fractal functions, such as the
Weierstrass type and 3-adic Cantor type, as intriguing examples within these
spaces and a few others. Employing prime elements, we develop a
Multi-Resolution Analysis (MRA) and examine wavelet expansions, focusing on the
orthogonality of both basic and fractal wavelet packets at various scales. We
utilize convolution theory to construct Haar wavelet packets and demonstrate
the orthogonality of all discussed wavelet packets within
$H^{\mathfrak{s}}(K_q)$, enhancing the analytical capabilities of these Sobolev
spaces.
