  Let $K$ be the function field of a smooth projective geometrically integral
curve over a finite extension of $\mathbb{Q}_p$. Following the works of Harari,
Scheiderer, Szamuely, Izquierdo, and Tian, we study the local-global and weak
approximation problems for homogeneous spaces of $\textrm{SL}_{n,K}$ with
geometric stabilizers extension of a group of multiplicative type by a
unipotent group. The tools used are arithmetic (local and global) duality
theorems in Galois cohomology, in combination with techniques similar to those
used by Harari, Szamuely, Colliot-Th\'el\`ene, Sansuc, and Skorobogatov. As a
consequence, we show that any finite abelian group is a Galois group over $K$,
rediscovering the positive answer to the abelian case of the inverse Galois
problem over $\mathbb{Q}_p(t)$. In the case where the curve is defined over a
higher-dimensional local field instead of a finite extension of $\mathbb{Q}_p$,
coarser results are also given.
