How do I explain to a six year old why people on the other side of the Earth don't fall off?

Today a friend's six year old sister asked me the question "why don't people on the other side of the earth fall off?". I tried to explain that the Earth is a huge sphere and there's a special force called "gravity" that tries to attract everything to the center of the Earth, but she doesn't seem to understand it. I also made some attempts using a globe, saying that "Up" and "Down" are all local perspective and people on the other side of the Earth feel they're on the top, but she still doesn't get it. How can I explain the concept of gravity to a six year old in a simple and meaningful way?