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We report a spectroscopic search for hypervelocity white dwarfs (WDs) that are runaways from Type Ia supernovae (SNe Ia) and related thermonuclear explosions. Candidates are selected from Gaia data with high tangential velocities and blue colors. We find six new runaways, including four stars with radial velocities (RVs) $>1000\,\rm km\,s^{-1}$ and total space velocities $\gtrsim 1300\,\rm km\,s^{-1}$. These are most likely the surviving donors from double-degenerate binaries in which the other WD exploded. The other two objects have lower minimum velocities, $\gtrsim 600\,\rm km\,s^{-1}$, and may have formed through a different mechanism, such as pure deflagration of a WD in a Type Iax supernova. The four fastest stars are hotter and smaller than the previously known "D$^6$ stars," with effective temperatures ranging from $\sim$20,000 to $\sim$130,000 K and radii of $\sim 0.02-0.10\,R_{\odot}$. Three of these have carbon-dominated atmospheres, and one has a helium-dominated atmosphere. Two stars have RVs of $-1694$ and $-2285\rm \,km\,s^{-1}$ -- the fastest systemic stellar RVs ever measured. Their inferred birth velocities, $\sim 2200-2500\,\rm km\,s^{-1}$, imply that both WDs in the progenitor binary had masses $>1.0\,M_{\odot}$. The high observed velocities suggest that a dominant fraction of the observed hypervelocity WD population comes from double-degenerate binaries whose total mass significantly exceeds the Chandrasekhar limit. However, the two nearest and faintest D$^6$ stars have the lowest velocities and masses, suggesting that observational selection effects favor rarer, higher-mass stars. A significant population of fainter low-mass runaways may still await discovery. We infer a birth rate of D$^6$ stars that is consistent with the SN Ia rate. The birth rate is poorly constrained, however, because the luminosities and lifetimes of $\rm D^6$ stars are uncertain.