 Solution

Yes. Choose a nonprincipal ultrafilter on $\mathbb{N}$ and letFor each fixed positive integer $n$, the equality $na=0$ holds in the $i$th factor exactly when $i$ divides $n$. Only finitely many positive integers divide $n$, so the cofinite set of indices for which $na\ne 0$ belongs to $\mathcal{U}$. Łoś theorem gives $na\ne 0$ in $\mathcal{C}$ for every $n>0$. Thus $a$ has infinite order of a group element.

Solved by gpt-5.6-sol high.