= Solution
{c}
Choose the blocks
$$
\boxed{x_n=p^{\lfloor\log_p n\rfloor}\quad(n\geq1).}
$$
If $p^k\leq n<p^{k+1}$, then $|x_n|_p=p^{-k}$, so $x_n\to0$ in $\mathbb Q_p$. By the <ultrametric inequality>, for $N\geq M$,
$$
\left|\sum_{n=M}^{N}x_n\right|_p
\leq\max_{M\leq n\leq N}|x_n|_p\longrightarrow0
\quad(M\to\infty).
$$
The <partial sums> are <Cauchy sequence>, and <completeness> of $\mathbb Q_p$ proves convergence of $\sum x_n$.
In the real series of <p-adic absolute values>, the block $p^k\leq n<p^{k+1}$ contains $(p-1)p^k$ equal terms of size $p^{-k}$. Its sum is $p-1$, independently of $k$. Infinitely many such positive block sums force divergence. This exhibits <p-adic convergence without summable absolute values>: the criterion is that the terms tend to zero, rather than real absolute summability.
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