There are different selection tools. There is an arrow that has a white fill (Selection Tool (A)) in it that behaves the way you say. But there is another arrow with a black fill (Selection Tool (V)) that allows selections of segments. Also the lasso tool will only select a portion of the line.
If the line you are trying to split up is already a symbol then the whole line will be selected in any scenario. You need to get down to the raw drawn object.