Could modern electronic chips (e.g. ARM and Intel processors) operate for millennia at sufficiently low temperatures and clock speeds?
I think the broader problem is that the chip is useless in isolation as it requires supporting components in order to make a useful circuit. Many of these components have a failure rate that exceeds that of silicon chips by many orders of magnitude.

In reliability engineering we know the failure rate of a serial system is greater than that of the highest failure rate component of the system. So to focus attention on what is arguably the most reliable component in a circuit is perhaps a priori but it is not germane to making hyper long life, functioning circuits for applications as proposed in the question.