When introducing $p$-adics, Kurt Hensel produced an incorrect proof of the transcendence of $e$: see https://mathoverflow.net/q/416296 for more details. The problem is that the proof relies on the "universality" of series, that is, the assumption that the series $$\sum_{n=1}^\infty\frac{p^n}{n!},$$ which converges in the $p$-adics, would converge to a number with similar properties as $e^p\in\mathbb R$. (This is false.)
My question is: is there a more dramatic way to apply this "proof" idea to show an obviously false result? For instance, using it to show the transcendence of a number that is actually algebraic.
In Example 6.2 here is a series of rational numbers converging to $0$ in $\mathbf R$ and to $1$ in $\mathbf Q_p$ for whichever prime $p$ you want. Example 9.5 there is an infinite series of rational numbers that converges to $8/7$ in $\mathbf R$ and to $-8/7$ in $\mathbf Q_3$ and $\mathbf Q_5$. Remark 9.7 gives an example of an infinite series of rational numbers that converges to $3$ in $\mathbf R$ and to $-3$ in $\mathbf Q_2$.