The $p$-adic sphere should just be $S(a,p^r)=a+p^{-r}\Bbb Z_p^\times$. If you combine this fact with the tree diagram interpretation of the $p$-adics this should help one visualize the boundary of balls. (Spheres are boundaries of balls in general metric spaces, right?)
Though I don't know that this applies to general ultrametric spaces.
@anon Thanks, Anon - I am not planning in going very far with general ultrametric spaces - I mainly want to know about the specific case of the $p$-adics - as I want to get to the understand why they are so useful in number theory. The topology thing is a bit of a side avenue, although it is interesting.
@OldJohn Though I mainly want to investigate, I think I can't resist teaching maths. I help many of my friends in math and I really enjoy teaching maths.
I had some papers from something I had been working on left on the table. And my niece asked me what it was for. I tried explaining that it was to calculate the between a function and a woobly surface. She looked oddly at me and asked why I ever wanted to figure that out.
"When determining the 5th reference to s causal relationship between x and xy squared, does one attempt clitoral reference or simply discard the model in number theory?"
