
Ilya's hat is called a Ushanka. It's characteristic of russian and mid-asia attire, commonly mongol and northern china.
Well, now that you have me thinking, I'm curious as to how I came to that conclusion also.
Let us see:

Ilya was first introduced to us in the first episode amid a very sad vocal. Even though she was smiling and giggling, the atmospher spoke of loneliness and death. I read these as someting that accompanied her everywhere, but I felt that it wasn't because she dealt death and sadness to those around her, but that it was the way she felt inside.
Whenever, I think of death, loneliness, and sadness, I think of the being alone, in a cold dark void, but here, the image of Ilya constructs a different portrait for me.
I was immediately aware and focused on her face and hair. Although I could not see her expression, the color white was stored in my memory. Her european origins was immediate apaprent with her white hair and fair skin. I won't say that other people's can't have teh same features, but it's not typical. Next is her purple color, a color I associate with death. In this case, it added further emphasis. Combined with the music and sounds, the image has a certain affect.
Instead of simply a cold dark void, an artic tundra amidst a winter storm came to my mind. The land is nearly always flat and with snow and ice all around. There's nothing around save the songs brought about by the wind and snow. It is a barren landscape. Standing there, I cannot help but feel despair as the cold creeps into my soul and my body becomes numb, unable to the recognize sensations of life.
From there, my thoughts move on to Russia, more precisely, a land I connect with Russia, Siberia. In the summers, the grasslands and tundras are an amazement of life. If ever I have the chance, I would like to ride across this piece of heaven, naturally, I'll be riding on a horse. But in the winter, it is another world altogether, a world I recognize as a wholly unpleasant place to be in or die in.

I could not clearly see her attire in whole at first, but i felt that her hat was familiar. Furthermore, she was wearing a scarf and a coat. This only reinforced the notion that she's form somewhere cold and forbidding. Her smile as well made me think of a princess. White > snow > serbia > princess.
But what's near serbia that could make me think of a princess? Anastasia. Thank you Disney.

It was right here that everything initially came together. I knew where her hat style is from. It was the last piece and the clearest indicator, I thought. She was definitely, russian, but most of my "evidence" were based on associations of thoughts pulled from images and emotions, nothing oncrete. That is until episode 4.

I don't know about you guys, but this facial expression and attire combination screams Russian.
Lastly,

No more doubt about it. She is a russian.