We report a study of the strain state of epitaxial MnSi films on Si(111) substrates in the thick film limit (100-500~Å) as a function of film thickness using polarization-dependent extended x-ray absorption fine structure (EXAFS). All films investigated are phase-pure and of high quality with a sharp interface between MnSi and Si. The investigated MnSi films are in a thickness regime where the magnetic transition temperature $T_\mathrm{c}$ assumes a thickness-independent enhanced value of $≥$43~K as compared with that of bulk MnSi, where $T_\mathrm{c} ≈ 29~{\rm K}$. A detailed refinement of the EXAFS data reveals that the Mn positions are unchanged, whereas the Si positions vary along the out-of-plane [111]-direction, alternating in orientation from unit cell to unit cell. Thus, for thick MnSi films, the unit cell volume is essentially that of bulk MnSi --- except in the vicinity of the interface with the Si substrate (thin film limit). In view of the enhanced magnetic transition temperature we conclude that the mere presence of the interface, and its specific characteristics, strongly affects the magnetic properties of the entire MnSi film, even far from the interface. Our analysis provides invaluable information about the local strain at the MnSi/Si(111) interface. The presented methodology of polarization dependent EXAFS can also be employed to investigate the local structure of other interesting interfaces. ; Comment: 11 pages, 10 figures