On detecting the X‐ray silhouette of a damped Lyα system We explore the possibility of resolving an image of a damped Lyα (DLA) system in absorption against an extended, diffuse, background X-ray source. Typical columns of neutral hydrogen in DLA systems are high enough to block out up to ~30% of the soft X-ray flux at an observed photon energy of 0.5 keV, and we find that ~1% of the area of extended X-ray sources at z ≳ 1 have their 0.5 keV flux reduced by at least 20% because of intervening DLA systems. We discuss the observability of such absorption and find that ≳300 photons per angular resolution element are required in the 0.3-8 keV band for its detection and in order to distinguish it from intrinsic surface brightness fluctuations. For the surface brightness of the currently known high-redshift extended X-ray sources, this requires an integration time of a few megaseconds on Chandra, if the maps are smoothed spatially to ≈2'' resolution. The exact required integration time depends on the DLA system's column density, metallicity, and, most strongly, its redshift. Current X-ray telescopes are likely to detect DLA systems with N < 10^22 cm^-2 only out to z ≈ 2.3. The availability of DLA systems with a suitably high column density for a silhouette detection is currently poorly known. We suggest that at low redshifts archival data of bright X-ray point sources may be useful in constraining the high-N end of the column density distribution. We briefly discuss an alternative strategy of searching for extended X-ray sources behind known DLA systems. Although with current X-ray telescopes the detections are challenging, they will be within the reach of a routine observation with a next generation X-ray telescope, such as the X-Ray Evolving Universe Spectrometer (XEUS) or Generation-X, and will deliver novel constraints on the nature of protogalaxies. 624 1 85-93 85-93 American Astronomical Society