I could use some help to jumpstart my intuition for lattices in vector spaces over non-Archimedean fields, like $\mathbb{Q}_p$ and $\mathbb{F}_q((t))$.

I have some intuition for $\mathbb{Z}$-lattices in $\mathbb{Q}$- or $\mathbb{R}$-vector spaces, though perhaps it's not sophisticated enough. The question formed while watching some talks about affine Grassmannians and Witt-vector analogs, realizing that my prior intuition was leading me astray.

I'd like to develop some intuition, in particular, for $R$-lattices in $K$-vectors spaces, where $R$ is $\mathbb{Z}_p$ or $\mathbb{F}_p[[t]]$ and $K$ is $\mathbb{Q}_p$ or $\mathbb{F}_p((t))$.

As a starting point, I would like to have a good feel for 2-dimensional lattices $R^2 \hookrightarrow K^2$.

**Question:** What is a good way to visualize these objects? Or what is a minimal list of properties that I should aim to capture in a mental image, especially to reflect similarities and differences to the case of $\mathbb{Z}$-lattices?

[Edit]: Details of current intuition:

My intuition for $\mathbb{Z}$-lattices mostly stems from the standard grid $\mathbb{Z}^2 \hookrightarrow \mathbb{R}^2$ and its images under $GL_2(\mathbb{R})$ action, moving the basis elements around arbitrarily, and then imagining higher dimensional analogs, to the extent possible.

I don't remember the full list of ways I noticed my intuition was leading me astray in the context of affine Grassmannians, but three insufficiencies with my current intuition are:

1) Unlike $\mathbb{Z} \subset \mathbb{Q}$, my mental image of $\mathbb{Z}_p \subset \mathbb{Q}_p$ already occupies two dimensions, rather than one, where I think about the $p^{n+1}\mathbb{Z}_p$ cosets as $p$ 2D blobs nested inside each $p^n\mathbb{Z}_p$ coset. So visualizing a rank-2 lattice here is akin to the difficulty of visualizing a plane curve over $\mathbb{C}$ using the intuition of plane curves over $\mathbb{R}$ - we've already got an ambient space of 4 dimensions.

2) $\mathbb{Z} \subset \mathbb{R}$ is cocompact, which is not the case for $\mathbb{Z}_p \subset \mathbb{Q}_p$. This is connected to the fact that if we move outward in $\mathbb{R}$, we keep encountering elements of $\mathbb{Z}$, whereas for my image of $\mathbb{Q}_p$, $\mathbb{Z}_p$ is located directly at the center, not occuring periodically.

3) My mental images for $\mathbb{Q}_p$ and $\mathbb{F}_p((t))$ are more or less the same, but I would like a mental image that somehow distinguishes them.

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