Yes, but the existence of such ultrafilters is a large cardinal hypothesis; it is equivalent to the existence of a measurable cardinal.

If each $A_i$ is countably infinite and $\cal U$ is countably complete ($\omega_1$-complete), then the ultraproduct $\Pi_i A_i/{\cal U}$ will be countable. To see this, fix enumerations of each $A_i$, and observe that any element $x\in\Pi_iA_i/{\cal U}$ is represented by some function $\vec x=\langle x_i\rangle_i$, where $x_i$ is the $n_i^{th}$ element of $A_i$. But by countable completeness, the value of $n_i$ must be constant on a set in $\cal U$, and so the function $n\mapsto [g_n]_{\cal U}$ where $g_n(i)$ is the $n^{th}$ element of $A_i$ is a bijection of $\omega$ with the ultraproduct. So it is countably infinite, as desired.

Meanwhile, it is important to note that every countably complete nonprincipal ultrafilter $\cal U$ on any set is actually $\kappa$-complete for some measurable cardinal $\kappa$, and so the hypothesis carries large cardinal strength.

A stronger result is possible, once you realize that any countably complete ultrafilter is actually $\kappa$-complete for a much larger cardinal. If $\cal U$ is a $\kappa$-complete ultrafilter on a set $I$ and the structures $A_i$ are uniformly bounded in size below $\kappa$, in a language of size less than $\kappa$, then the ultrapower $\Pi_i A_i/{\cal U}$ is isomorphic to one of the $A_i$. The reason is that we may assume $\kappa$ is a measurable cardinal, and in particular, it is also inaccessible. And so there are fewer than $\kappa$ many structures in that language of size at most $\delta$, for any fixed $\delta<\kappa$. It follows by $\kappa$-completeness that the measure $\cal U$ must concentrate on a set of indices $i$ for which the $A_i$ are all isomorphic to each other. And it follows by the reasoning in the first argument that this common $A_i$ is also isomorphic to the ultrapower $\Pi_i A_i/{\cal U}$.

The result mentioned in the previous paragraph is the size $\kappa$ analogue of the fact for ultrafilters $\cal U$ on $\omega$ that the ultrapower $\Pi_i A_i/{\cal U}$ of structures $A_i$ having uniformly bounded finite size in a finite language is isomorphic to one of them, namely, to the one whose isomorphism type occurs on a set in ${\cal U}$.