p-adic convergence without summable absolute values ($x_n=p^{\lfloor {\log }_{p}n\rfloor }$)

In a complete Non-Archimedean absolute value field, a series converges if and only if its terms tend to zero: the ultrametric inequality bounds every finite tail by its largest term. The displayed sequence tends to zero in ${\mathbb{Q}}_p$. But each block $p^k\le n<p^{k+1}$ contributes $p-1$ to the real sum of its p-adic absolute values, so that real sum diverges.