Both surrounded by and surrounded with share the semantics 'to encircle', but each means much more than that.
Consider this example, an army surrounds the castle. Most readers would assume given the word choice that the castle is under seige; however, given a larger context, say, that sentence was part of the description of a day of celebration, the assumption would be soldiers on parade. My point is, without context we don't really know which meaning it is; however, if we add 'by' or 'with', all of a sudden the scope of the semantics becomes narrower:
Which army surrounds the castle, the warring side's or the castle's?
 The castle was surrounded with an army.
<supported as in, say, protected; i.e., the castle's army>
 The castle was surrounded by an army.
<supported as in, say, enclosed; i.e., the warring side's army>
He was surrounded with our love. <protected>
He was surrounded by our love. <enclosed>
Note, 'enclosed' isn't the right word for that example, but it's close enough to get my point across. Maybe you, Clark, can provide the word.
The garden was surrounded with a fence. <protected>
The garden was surrounded by a fence. <enclosed>
Oh, and so to answer your question,
Ex: We were surrounded by moutains <enclosed by; the meaning 'protected' doesn't come into play>