Without further context, I would assume that the girl is not seeing anyone at the moment (she is not in a relationship), and she is looking for someone to be in a relationship with.
On the other hand, she may indeed be telling you (on the phone for instance) that she is out shopping. (Thank you Edwin Ashworth!)
When we use the phrase on the market we are contrasting the idea that something is available to buy with the notion that it might not be available to buy:
Are they on the market yet?
When we use a preposition phrase with in after the word leader(s), we are indicating the area that the leader is leader in:
He was a leader in the field of nuclear electromagnetic resonance.
Deep can mean "A long way inside" (sense 6). When you refer to "the mountains" (plural) you mean a mountainous region. Hence this means "A long way inside a mountainous region". It doesn't mean underground, nor does it necessarily mean on a mountain top. It does suggest you are far from civilisation.
Best Answer
You may want to have a look at this question.
Without further context, I would assume that the girl is not seeing anyone at the moment (she is not in a relationship), and she is looking for someone to be in a relationship with.
On the other hand, she may indeed be telling you (on the phone for instance) that she is out shopping. (Thank you Edwin Ashworth!)