The connection between Gauss sums and the $p$-adic Gamma function is a very deep theorem, and is known as given by the Gross-Koblitz formula.
Here is the original paper by Gross and Koblitz: Gauss sums and the $p$-adic $\Gamma$-function. (1979)
For some more specific cases and applications, see the book by Berndt, Evans, and Williams, "Gauss and Jacobi Sums. (1998)"