Hi guys, thanks for the ideas. Right now I'm leaning towards the putting her in the freezer. I could make her a little fleece blankie and wrap her in it, then put her in a ziplock bag and then in a little container. I know the blankie wouldn't really keep her warm, but at least I'd feel a little better about it. We buried her friend, Emmy, right by a lake when she died and we could take Lily to the lake and bury her by Emmy when the ground thaws.
When I was little and my first mouse died, my dad took a distraught me to the store and bought me a small glass Corningware dish with a plastic lid, some pretty tissue paper with hearts on it and a gold paint pen. I wrapped the mouse in the paper, wrote all over the container with the pen and then we buried him in the backyard. We did the same with all my other little pets that died when I lived there and when I was out of the country and my hamster died, my dad buried her in the same fashion. I think I'll get the same type of dish and do it with Lily!
Thank you for your support and good ideas!