  We study $p$-adic Euler's series $E_p(t) = \sum_{k=0}^{\infty}k!t^k$ at a
point $p^a$, $a \in \mathbb{Z}_{\ge 1}$, and use Pad\'e approximations to prove
a lower bound for the $p$-adic absolute value of the expression $cE_p\left(\pm
p^a\right)-d$, where $c, d \in \mathbb{Z}$. It is interesting that the same
Pad\'e polynomials which $p$-adically converge to $E_p(t)$, approach the Hardy
integral $\mathcal{H}(t) = \int_{0}^{\infty} \frac{e^{-s}}{1-ts}ds$ on the
Archimedean side. This connection is used with a trick of analytic continuation
when deducing an Archimedean bound for the numerator Pad\'e polynomial needed
in the derivation of the lower bound for $|cE_p\left(\pm p^a\right)-d|_p$.
Furthermore, we present an interconnection between $E(t)$ and $\mathcal{H}(t)$
via continued fractions.
