We know that $\nabla •\textbf{E(r)}= ρ(\textbf{r})$.
However, consider a finite volume with volumetric charge embedded in empty space. Since the electric field will decrease with distance from the source, its divergence should be non zero in the empty space.
Yet there's no charge in the empty space, so $\rho$ would be zero, and so would divergence.
What am I missing?
The divergence of a vector field $E$ at a point $p$ is determined by the behaviour of $E$ in the neighbourhoods of $p$.
Choose a neighbourhood $U$ of $p$ (it can be as small as you want), and consider the field lines of $E$ in $U$. Since the field is weaker away from the charges, some field lines are going to be "stronger" than others. The divergence of $E$ at $p$ measures the lines that concentrate towards $p$ minus the lines that point outwards $p$.
In this case, the lines that concentrate towards $p$ are coming from the charges, so they are stronger than the lines that point away from $p$. But the lines that point away are more abundant (i. e. they cover a wider volume) than the stronger lines. This is easier to see in the case of a field generated by a point charge $q$:
In this picture, the yellow lines are field lines, and the blue circle is the neighbourhood $U$. The grey line separates the lines closer to the charge (they are stronger, and some of them are pointing towards $p$) and the ones away from $q$ (all of them point outwards from $p$, and they cover a larger part of $U$, but they are weaker).
The general case is pretty much obtained by adding up (integrating) the fields spanned by infinite point charges, but divergence is linear, so the divergence of the sum will be equal to the sum of the divergences (which are zero).
The field decreases, that's true, but the area of the imaginary sphere around the charge grows, so the to total flux of electric field remains the same. Hence, the divergence is zero, in agreement with Gauss' Law.
Just because the electro-static field decreases with the distance from the source doesn't mean that its divergence can't be zero.
For example, consider the Coulomb field. It can be shown that the divergence is equal to zero outside from the point-like source. Inside the source point, it goes to infinity, much like $\rho$.
