A counterexample may be interesting. Consider the sequence
$$ a_n = \frac{n!}{n! + 1} $$
In the real numbers, we obviously have
$$ \lim_{n \to +\infty} a_n = 1, $$
but in every $p$-adic field, we have
$$ \lim_{n \to +\infty} a_n = 0, $$
so you should be wary of the idea of trying to sum a series of real numbers by transplanting it to the $p$-adics.
One thing you can consider is $\mathbb{Q}((x))$, the field of rational (formal) Laurent series. In this field, you have an identity
$$ \sum_{n=0}^{+\infty} x^n = \frac{1}{1-x}. $$
There is no issue of convergence or anything here; you just check that multiplying the left hand side by $1-x$ gives you $1$.
There is a subfield of $\mathbb{Q}((x))$ that consists of only those Laurent series for which replacing $x$ by $2$ yields a convergent $2$-adic sum. Evaluation at $2$ then becomes a field homomorphism to the $2$-adic numbers. Since $\sum_{n=0}^{+\infty} x^n$ is in that subfield, its image in $\mathbb{Q}_2$ must be the same as the image of $1/(1-x)$: i.e. $-1$.
Wikipedia has a page on divergent series which talks about "summation methods". You may find this another useful starting point.