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I've seen this happen in an HP42S. The fault is either a short in one of the chips (which is not easily repairable unless you have another 11C to take parts from), or more likely a leaky capacitor across the battery. In the Voyager series, this is the only tantalum bead capacitor on the board (either on the flexible PCB on the original series, or on the only PCB in the later series). In the Pioneers, it's a surface mount capacitor near the battery contacts. I think it's something like 10uF or 22uF
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With all due respect to Tony, I've never seen a leaky battery backup capacitor. Easy enough to swap out but my money would be on a bad processor chip.
If you have the two chip version 11C, you're in luck as you can use the processor from a 12C. Any made in USA 12C will have the cpu you need (1LF5-0301). The later 1LM2-0001 will work as well.
If you have a later model (serial starts with 28), you might be out of luck as they changed to a single chip version (1LQ9-0321). They're rather rare and the only source would be from an identical age 11C.