  Let $p$ be any prime and let $a$ and $n$ be positive integers with $p\nmid
n$. We show that $$\sum_{k=1}^{p^a-1}\frac{B_k}{(-n)^k}\equiv
a(-1)^{n-1}D_{n-1}\pmod {p},$$ where $B_0,B_1,\ldots$ are the Bell numbers and
$D_0,D_1,\ldots$ are the derangement numbers. This extends a result of Sun and
Zagier published in 2011. Furthermore, we prove that
$$(-x)^n\sum_{k=1}^{p^a-1}\frac{B_k(x)}{(-n)^k}\equiv
-\sum_{r=1}^ax^{p^r}\sum_{k=0}^{n-1}\frac{(n-1)!}{k!}(-x)^k\pmod{p\mathbb
Z_p[x]},$$ where $B_k(x)=\sum_{l=0}^kS(k,l)x^l$ is the Bell polynomial of
degree $k$ with $S(k,l)\ (0\le l\le k)$ the Stirling numbers of the second
kind, and $\mathbb Z_p$ is the ring of all $p$-adic integers.
