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The goal of this article is to associate a $p$-adic analytic function to the Euler constants ${\gamma }_p(a,F)$, study the properties of these functions in the neighborhood of $s=1$ and introduce a $p$-adic analogue of the infinite sum ${\sum }_{n\ge 1}f\left(n\right)/n$ for an algebraic valued, periodic function $f$. After this, we prove the theorem of Baker, Birch and Wirsing in this setup and discuss irrationality results associated to $p$-adic Euler constants generalising the earlier known results in this direction. Finally, we define and prove certain...