Let $R$ be a discrete valuation ring qith quotient field $Q$ and let $t\in R$ be a generator of the unique maximal ideal in $R$. Let $V$ be a finite-dimensional $Q$-vector space. Then one can consider the set of all homothety classes of $R$-lattices in $Q$ (i.e. finitely generated $R$-submodules of the same rank).

The affine building is the simplicial complex whose vertices are homothety classes of lattices (meaning $L\sim \lambda L$ for $\lambda\in Q\setminus \{0\}$) and a set $\{[L_0],\ldots ,[L_m]\}$ of homothety classes spans a simplex if there are represenatatives $L_0,\ldots,L_n$ with $L_0\subset L_1\subset \ldots \subset L_m\subset t^{-1}L_0$.\

I read several times that there is a Euclidean structure on this building that turns it into a CAT(0)-space but I never read explicitly what the metric on the simplex is.

In general for any maximal simplex (aka chamber) one can find a basis $e_1,\ldots,e_n$ of $V$ such that $e_1,\ldots,e_i, te_{i+1},\ldots te_n$ is an $R$-basis for $L_i$ (for $i=0,\ldots, n-1$).

So the question is: what is the metric on this simplex. My guess is that the lattice $L_i$ corresponds to the point $(0,\ldots,0,1,\ldots,1)\in \mathbb{R}^n$ (with $i$ zeros) and that its homothety class corresponds to the orthogonal projection of this point onto the plane $\{x\in \mathbb{R}^n| \sum_i x_i=0\}$.

Then we can take the convex hull of those $n$- points in that plane to get an Euclidean simplex which gives the building the structure of an Euclidean simplicial complex.

However I could not verify that it is a CAT(0) space. One would have to show that the links are CAT(1) (with the induced spherical metric) and I do not have a clue how this works. However I had the impression that this is rather well known and I just don't see it.