$p$-adic Fourier transforms and orthogonality relations

Question

1. In $\mathbb{C}$, we have the following orthogonality relation $$ \int_{0}^{1} e^{2\pi i (m-n)x} dx = \begin{cases} 1 & \mbox{ if } m = n;\\ 0 & \mbox{ otherwise.} \end{cases} $$ Do we have something similar for $\mathbb{C}_p$ with $e^{2\pi i (m-n)x}$ replaced with a similar $p$-adic valued function?

2. Is there an analogue of the Fourier transform (continuous or discrete cyclic or linear) for $\mathbb{C}_p$? Thanks