How to power a 10-watt electric circuit without grid connection in cold countries?
You want electricity. What you want it for is kind of irrelevant. You want it, and for your own reasons, you're looking at solar as a starting point.

Solar is lovely when space is plentiful and light isn't a problem. But storage of electricity isn't reliable at the temperatures you're looking at. So other options are: temperature controlled environment for the batteries or generate it on demand.

Generating on site is the easiest method. After all, that's what is done in out of the way places such as here. 

But that has it's own problems, costs and environmental issues.

Or you could do wind generation, which is often able to give you more energy per unit cost than solar (YMMV). But again, you're limited to when and where the wind is blowing.

Don't underestimate controlling the environment of the batteries. They generate heat when current is flowing, in or out. They are resistors in a way. So, turn that heat to your advantage. You haven't supplied all the numbers for current, voltage and power consumption. But the batteries will generate heat when used, not enough to keep them warm at ex teams, but it's something. Here's an idea: make a very will insulated box, put a small heater set to keep the box at +5C. Power the heater from some cheap fossil fuel, or, if you can be confident of the power from wind, perhaps that will be enough to power an electric heater. Clearly that depends on your set up, requirements and location etc.