Early microprocessors often used NMOS or PMOS transistor technology (see this question for their use in early Intel chips). Techniques such as implementing registers with dynamic memory cells (instead of static logic) and precharging the buses were often used to save on the transistor counts. However, such techniques meant that the processor could not tolerate a stopped clock signal, as charge would eventually drain away.
In contrast, CMOS processors are more likely tolerate a stopped clock. The RCA 1802 is a good example; the clock may be stopped indefinitely. The PowerPC 750 was CMOS and dynamic, but its radiation-hardened version RAD750 is CMOS and static.
Were there any NMOS/PMOS processors that would still function properly after the clock was stopped and restarted?