We study the collection of finite elements $��_{1}(\mathcal{U}(E,F))$ in the vector lattice $\mathcal{U}(E,F)$ of orthogonally additive, order bounded (called abstract Uryson) operators between two vector lattices $E$ and $F$, where $F$ is Dedekind complete. In particular, for an atomic vector lattice $E$ it is proved that for a finite element in $��\in \mathcal{U}(E,\mathbb{R})$ there is only a finite set of mutually disjoint atoms, where $��$ does not vanish and, for an atomless vector lattice the zero-vector is the only finite element in the band of $��$-laterally continuous abstract Uryson functionals. We also describe the ideal $��_{1}(\mathcal{U}(\mathbb{R}^n,\mathbb{R}^m))$ for $n,m\in\mathbb{N}$ and consider rank one operators to be finite elements in $\mathcal{U}(E,F)$.