Vector spaces over fields are pseudofinite, and this remains true for vector spaces over division rings that are finite-dimensional over their center. We also construct a division ring such that the nontrivial vector spaces over it are not pseudofinite, using Richard Thompson's group F. The idea behind the construction comes from a first-order axiomatization of the class of division rings all whose nontrivial vector spaces are pseudofinite.
"Division rings whose vector spaces are pseudofinite." J. Symbolic Logic 75 (3) 1087 - 1090, September 2010. https://doi.org/10.2178/jsl/1278682217