But the space between us actually does increase, if you are confined to that surface. In general relativity theory the same thing happens. If we don't move, then a black hole is placed in between us, the distance between us is vastly increases. To get from me to you it would have to go way down into that space. It is a huge amount of space crowded in around the black hole.
Space is a flexible tangible thing in modern physics.
Now I talk about teaching this to kids. When I was six, I was sort of mystified by this idea that space was so fixed, I mean the distances between things seemed to change every year. The hills got smaller, the apartment buildings got smaller. Everything was relative to me. I had to be taught that space was fixed and that I was growing.