= Solution
The <Schwartz-Bruhat space> $\mathcal S(F)$ of a non-Archimedean <local field> $F$ is the vector space of locally constant, compactly supported complex-valued functions on $F$. Fix a nontrivial continuous <additive character> $\psi:F\to\mathbb C^\times$ and a <Haar measure> $dx$. With the sign convention required in the question, the <Fourier transform over a local field> is
$$
\widehat f(y)=\int_F f(x)\psi(xy)\,dx.
$$
Let
$$
\mathcal O_F^\perp=\{y\in F:\psi(xy)=1\text{ for every }x\in\mathcal O_F\}
$$
be the <annihilator of the valuation ring>. Translation invariance gives
$$
\widehat{\mathbf1_{\mathcal O_F}}(y)
=\int_{\mathcal O_F}\psi(xy)\,dx
=\operatorname{vol}(\mathcal O_F)\mathbf1_{\mathcal O_F^\perp}(y).
$$
Indeed, the integral is the volume when the character is trivial; otherwise translation by an element on which the character is nontrivial multiplies the integral by a scalar different from one, forcing it to vanish. With the usual character of conductor $\mathcal O_F$ and the normalization $\operatorname{vol}(\mathcal O_F)=1$, this becomes
$$
\widehat{\mathbf1_{\mathcal O_F}}=\mathbf1_{\mathcal O_F}.
$$
For the canonical character induced from $\mathbb Q_p$, the annihilator is instead the <inverse different> and the displayed general formula applies.
If $g(x)=f(ax+b)$ with $a\ne0$, the substitution $u=ax+b$ and the scaling rule $dx=|a|^{-1}du$ give
$$
\begin{aligned}
\widehat g(y)
&=\int_Ff(ax+b)\psi(xy)\,dx\\
&=|a|^{-1}\int_Ff(u)\psi((u-b)y/a)\,du\\
&=\psi(-by/a)|a|^{-1}\widehat f(y/a).
\end{aligned}
$$
Every locally constant compactly supported function is a finite linear combination of characteristic functions of cosets $b+a\mathcal O_F$: compactness extracts finitely many cosets on which the function is constant. The formula just proved, together with the transform of $\mathbf1_{\mathcal O_F}$, shows that the transform of each such characteristic function is again locally constant and compactly supported. Therefore the <Fourier transform over a local field> maps $\mathcal S(F)$ to itself.
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