I think that we can say is a finite ring. The only way it could be infinite is if the group of units, call it , were infinite. But then for any with would be an infinite number of non-unital elements, which is a contradiction.
So I guess we must assume that there is at least one non-zero, non-unital element, otherwise an infinite ring of nothing but units with an additive identity thrown in (such as ) would destroy the argument.
In short: yes, unless is a field!