Lines in a p-adic plane The geometry of lines in $\mathbb{R}^2$ is fundamental to mathematics and likewise for lines in $\mathbb{C}^2$ since $\mathbb{C}^2 \cong \mathbb{R}^4$. But is there a good treatment of lines in $\mathbb{Q}_p^2$, i.e. the $p$-adic plane? 
For example, suppose $y= \zeta x$ is a $p$-adic line. How does it interact with the lattice $\mathbb{Z}^2 \subset \mathbb{Q}_p^2$? Certainly  the line will pass through a well-determined set of lattice points if and only if $\zeta \in \mathbb{Q} \subset \mathbb{Q}_p$. So what if $\zeta \not\in \mathbb{Q}$? Can we make a statement (similar to irrational slopes in the reals) that the line passes arbitrarily close to lattice points? 
Perhaps more importantly, is there any value added by thinking geometrically since most of these questions could be answered algebraically and the non-archimedean space is difficult to imagine?

Unfortunately, there are no lattices in $p$-adic vector spaces. The reason is that if $v$ is a nonzero vector, then $p^nv$ approaches $0$ in the $p$-adic topology, so $0$ is an accumulation point of every subgroup.
