The ~166,000 km2 fan-shaped deposit at Arsia Mons contains three characteristic facies (ridged, knobby, and smooth), which are interpreted as the depositional remains of a cold-based glacier that was present on the west--northwestern flanks in the Late Amazonian period of Mars history. Here, we consider several high-elevation graben on the western flank of Arsia Mons that are interpreted as the source regions for late-stage, cold-based glaciers that overflowed graben walls, advanced tens to hundreds of kilometers downslope, experienced subsequent retreat, and left distinctive depositional features similar to those associated with cold-based glaciers in the Dry Valleys of Antarctica. These new observations provide additional support for a cold-based glacial interpretation of the Tharsis Montes fan-shaped deposits. Morphological evidence suggests that the largest of these graben was the primary source region for the most proximal smooth facies lobes at Arsia Mons, and map-plane ice sheet reconstructions are consistent with these observations. This relationship has significant implications for the relative ages of the individual facies within the fan-shaped deposit, including the possibility that the smooth facies represents several distinct phases of glaciation. The high relief and large elevation differences between the upslope and downslope walls of these graben appear to be critical factors for ice accumulation and the generation/collection of rockfall within glacial accumulation zones. We suggest that rockfall was the dominant process responsible for debris-cover formation on the glaciers sourced within these graben and consider the implications for the entire Arsia fan-shaped deposit. New crater count data suggest that the fill material in the large Arsia graben formed within the past ~100 Myr (model age of ~65 Ma). Taken together, these new observations provide additional evidence for several periods of Late Amazonian climate change on Mars within the past few 100 Myr resulting in episodes of tropical mountain glaciation. Finally, MOLA topography reveals that several lobate features interpreted as remnant debris-covered ice from the most recent phase of glaciation are presently hundreds of meters thick, suggesting the possibility of long-term, near-surface water ice survival in the equatorial regions of Mars.