We assess the information that HST observations of stellar Lyα lines can provide on the heliosheath, the region of the heliosphere between the termination shock and heliopause. To search for evidence of heliosheath absorption, we conduct a systematic inspection of stellar Lyα lines reconstructed after correcting for ISM absorption (and heliospheric/astrospheric absorption, if present). Most of the stellar lines are well centered on the stellar radial velocity, as expected, but the three lines of sight with the most downwind orientations relative to the ISM flow (χ1 Ori, HD 28205, and HD 28568) have significantly blueshifted Lyα lines. Since it is in downwind directions that heliosheath absorption should be strongest, the blueshifts are almost certainly caused by previously undetected heliosheath absorption. We make an initial comparison between the heliosheath absorption and the predictions of a pair of heliospheric models. A model with a complex multicomponent treatment of plasma within the heliosphere predicts less absorption than a model with a simple single-fluid treatment, which leads to better agreement with the data. Finally, we find that nonplanetary energetic neutral atom (ENA) fluxes measured by the ASPERA-3 instrument on board Mars Express, which have been interpreted as being from the heliosheath, are probably too high to be consistent with the relative lack of heliosheath absorption seen by HST. This would argue for a local interplanetary source for these ENAs instead of a heliosheath source.