As a native speaker of British English (or real-english to be obtuse) I would agree with both the posts of Vality and Gilles but would note, specifically in the run example, that the reference between little and short has been missed.
I would say "I went for a little run" or more likely I went for a "short run" as this is denoting either distance or time.
People can be 'short' denoting height e.g. "he was too short to ride the roller-coaster"; 'little' denoting age or frame i.e. a "little old lady" again, may be endearing; or 'small' e.g. "he was only small" again referring to frame, perhaps a little more 'frank' - i.e. more plain speaking than to refer to someone as 'little'.
The quote you mention refers to a child, thus the use of 'little' makes sense as a term of endearment.
edit: I would also add that in your example 'She's still very small' you've introduced a concept of time. If you said 'She's very small' that would be fine. It feels odd that you've added 'still' as this suggests that she should be something different. i.e. Are you expecting her to be bigger, taller or larger?
NOTE: I would have added this as a comment, but do not currently have the rep to post as such...