“das ganze Jahr über” belongs together. It is called a “temporale Präpositionalgruppe”. Read more: Präpositionen - mein-deutschbuch.de
Subjekt = Schnee und Eis
Prädikat = liegt
Objekt (mit Präposition) = Auf den Bergen
Other parts are not needed to build a sentence, but they can be added.
Like Prinz_of_May said the “über” belongs to the “das ganze Jahr”. And as I said is “Schnee und Eis” the subject in this sentence.
When you read more, you’ll know that “über” doesn’t have to come before the duration of time indicated. Very often, it’s after.
I admit that word order and order of parts of the sentences in German is on the one hand very flexible and follow some rules on the other hands. They are easy for native speakers but they can become tricky for German learners. The only thing that helps is to read a lot to get used to it. The simpliest way to build a sentence is Subjekt - Prädikat - Objekt, but it can also be the other way round like in this case.
I get what you are saying, but there is nothing stopping über from belonging to Schnee und Eis either. From a grammatical standpoint, what I am saying is not wrong and depending on the context also brings out additional meaning that is not there using your interpretation.
Folks, I had no idea that I was unleashing such a discussion but for me as a new learner of German, it has been most educational. I enjoy learning and I enjoy learning from a number of perspectives so this discussion is especially gratifying from point of view. I must tell you a funny story that illustrates how language is used in America where I am suppose to understand my native tongue. Over 40 years ago I moved into the foothills of the Applachian Mountains, a beautiful area of the US to live in with a superb climate, but as nice as it is, my problems began almost immediately when I started speaking and I asked someone for some help and they responded to me…“I don’t care to help you.” Anyone of you think you know what that means?
The only way I would understand that is: “I don’t want to help you”.
But I’m guessing it has some other meaning in hillbilly county, right? 
Yes, you are right Prinz_of_May. I was accosted many times with phrases containing these words "You don’t care to “do something”, “If you don’t care…” and so on before it became apparent that this was really an eager offering to help me do something. They meant and thought they said to me…“I am glad to help you.” “I do not mind to help you at all.”----it just did not sound that way to my ears. Ha!
I suppose a person’s tone of voice and manner can also help to clarify the meaning in these kinds of situations?
I guess, in a real life context, one would get the correct picture most of the time?
Actually, if you had heard what I heard that first time, it sounded very much like they were not interested in helping at all so this was a stumper for me and I had to ask someone what it meant because it really did not sound good—I am glad I was wrong and the people here are very helpful in nature—and funny too, Ha!
Good ol’ App-a-latch-a!
Since we are all about pronunciation here…App a lach ua would be rendered in our neck of the woods…
App-uh–latch-oh-were . Ha!
There is a county in Florida near Gainesville, Fl and the University of Florida (better known by many as “The Swamp” or “Gator Country” with the same spiritual heritage…its name is printed Alachua and pronounced by the born and reared there as
Uh-lach-oh-were County…another useless tidbit—enjoy.
We hillbillies are every where—like the Shadow!
Virtually everyone else in the States pronounces it with LAY in the middle.
Wasn’t until I watched a documentary on the dialect from around there a few years ago that I learned they pronounce it with a LATCH in the middle.