When someone says: "I'm deep in the mountains", does it mean that he is on a high mountain? because I don't think that going inside a mountain is possible.
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.