Abstract Space-bounded one-way cellular language acceptors (OCA) are investigated. The only inclusion known to be strict in their time hierarchy from real-time to exponential-time is between real-time and linear-time! We show the surprising result that there exists an infinite hierarchy of properly included OCA-language families in that range. A generalization of a method in Terrier (Theoret. Comput. Sci. 156 (1–2) (1996) 281) is shown which provides a tool for proving that languages are not acceptable by OCAs with small time bounds. The hierarchies are established by such a language and a translation result. In addition, a notion of constructibility for CAs is introduced, along with some of its properties. We prove several closure properties of the families in the hierarchy.