Ok. I live off the grid, but I don't have experience with solar for the coop because we are building the coop as soon as the snow melts. That means another month where I am. We have frigid winters and humid summers and it can get quite hot in the summer months.
Having a solar panel charge a battery to use for heat will not work very well. When heat is created with electricity it's called a controlled dead short. That is very inefficient. Also- you need a deep cycle battery and sufficient charging capacity.
If you use some sort of hot/warm air panel or maybe even hot water panel. I'm not sure how cold it gets where you are, but heliodyne has hot water panels that can tolerate freezing. It wouldn't make it a toasty 70 degrees, but it might take some of the chill off. I've never used hot water panels for heat (only for heating domestic hot water), so I have no idea. If you go the hot water route, you need power for a circulating pump which would be more efficient than electric heat. Geothermal would be ideal, but again you're dealing with circulating pumps. This is a chicken coop. The expense of all the above would be horrendous for a 10x10 space. If you find a small warm air system, link it here, I'd be interested in checking it out.
Best bet: Follow the advice of the person above. Heat may be a bigger issue than cold. Insulate well, seal junctions, add lots of vents that won't blow on the birds in winter. Maybe utilize a passive solar design like they did in the old days: Put the coop in the sun and plant tall shrubs or trees around it. Deciduous trees/shrubs will shed their leaves in winter and let the sun shine on it to allow for solar gain. You can even buy solar gain window samples at a building supply place. In the summer, the bush/tree will have leaves on it and will shade the coop in the summer, minimizing solar gain.
I am going to use tin for roofing, and place a little removable tin awning over each window to reduce drafts and shade it a little in the summer. If the coop is well insulated, that should help with the heat in the summer too. I will also check before I plant said shrubs to be sure they aren't poisonous to the chickens.
Using solar for lighting is easy-peasy and can be quite efficient, that's what we're doing. Solar LED flood lights. I'm hoping that does something to dscourage predators too.
As also said above, it might just be better for you to bury a line out to the coop.
Good luck, and let me know what route you go, I'll be interested to see how it works.