  Type Ia and other peculiar supernovae (SNe) are thought to originate from the
thermonuclear explosions of white dwarfs (WDs). Some of the proposed channels
involve the ejection of a partly exploded WD (e.g. Iax SN remnant) or the
companion of an exploding WD at extremely high velocities ($>400$
km\,s$^{-1}$). Characterisation of such hyper-runaway/hypervelocity (HVS) WDs
might therefore shed light on the physics and origins of SNe. Here we analyse
the \textit{Gaia} DR3 data to search for HVS WDs candidates, and peculiar
sub-main-sequence (sub-MS) objects. We retrieve the previously identified HVSs,
and find 46 new HVS candidates. Among these we identify two new unbound WDs and
two new unbound sub-MS candidates. The remaining stars are hyper-runaway WDs
and hyper-runaway sub-MS stars. The numbers and properties of the HVS WD and
sub-MS candidates suggest that extreme velocity ejections ($>1000$
km\,s$^{-1}$) can accompany at most a small fraction of type Ia SNe,
disfavouring a significant contribution of the D6-scenario to the origin of Ia
SNe. The rate of HVS ejections following the hybrid WD reverse-detonation
channel could be consistent with the identified HVSs. The numbers of
lower-velocity HVS WDs could be consistent with type Iax SNe origin and/or
contribution from dynamical encounters. We also searched for HVS WDs related to
known SN remnants, but identified only one such candidate.
