I saw the following identity few times while studying the subject of p-adic analysis, but I struggle a little bit trying to prove it. the identity is as follows $\sum_{n=0}^\infty(-1)^{n+1}\frac{x^n}{n!}a_n=\frac{1-x^p}{1-x}\exp_p(x+\frac{x^p}{p})$ where $\Gamma_p(x+1)=\sum_{n=0}^\infty\binom{x}{n}a_n$ is the Mahler series of $\Gamma_p(x+1)$
I tried using the integral defition of the Gamma function and Mahler's theorem but I didn't figured it out yet. Any suggestions?
Did you look at the treatment of the $p$-adic Gamma function in Lang's book Cyclotomic Fields I and II? See Chapter 14, especially the appendix "Barsky's Existence Proof for the $p$-adic Gamma Function". It is based on Barsky's paper "On Morita's $p$-adic gamma function," Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 1, 23–27.
By the way, I advise writing the exponential term on the right side as $\exp(x + x^p/p)$, not $\exp_p(x + x^p/p)$ The equation you ask about is a formal power series identity, so there is no purpose in writing $\exp_p$ instead of $\exp$. You're just working with the exponential power series with $x+x^p/p$ used in place of $x$.