We have used the Very Long Baseline Array to image 18 quasars with foreground damped Lyman-$alpha$ systems (DLAs) at 327, 610 or 1420 MHz, to measure the covering factor $f$ of each DLA at or near its redshifted HI 21cm line frequency. Including six systems from the literature, we find that none of 24 DLAs at $0.09 < z < 3.45$ has an exceptionally low covering factor, with $f sim 0.45 - 1$ for the 14 DLAs at $z > 1.5$, $f sim 0.41 - 1$ for the 10 systems at $z < 1$, and consistent covering factor distributions in the two sub-samples. The observed paucity of detections of HI 21cm absorption in high-$z$ DLAs thus cannot be explained by low covering factors and is instead likely to arise due to a larger fraction of warm HI in these absorbers.
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