Splitting and making explicit the de Rham complex of the Drinfeld space

Let $p$ be a prime number, $K$ a finite extension of ${\mathbb{Q}}_p$ and $n$ an integer $\ge 2$. We completely and explicitly describe the global sections ${\mathrm{\Omega }}^{\bullet }$ of the de Rham complex of the Drinfeld space over $K$ in dimension $n-1$ as a complex of (duals of) locally $K$-analytic representations of ${\mathrm{G}\mathrm{L}}_n(K)$. Using this description, we construct an explicit section in the derived category of (duals of) finite length admissible locally $K$-analytic representations of ${\mathrm{G}\mathrm{L}}_n(K)$ to the canonical morphism of complexes ${\mathrm{\Omega }}^{\bullet }\twoheadrightarrow H^{n-1}({\mathrm{\Omega }}^{\bullet })[-(n-1)]$.