  The multi-variable Schmidt polynomials are defined by $$
S_n^{(r)}(x_0,\ldots,x_n):=\sum_{k=0}^n {n+k \choose 2k}^{r}{2k\choose k} x_k.
$$ We prove that, for any positive integers $m$, $n$, $r$, and $\varepsilon=\pm
1$, all the coefficients in the polynomial $$
\sum_{k=0}^{n-1}\varepsilon^k(2k+1) S_k^{(r)}(x_0,\ldots,x_k)^m $$ are
multiples of $n$. This generalizes a recent result of Pan on the divisibility
of sums of Ap\'ery polynomials.
