  Massive D=4 N=2 supersymmetric sigma models typically admit domain wall
(Q-kink) solutions and string (Q-lump) solutions, both preserving 1/2
supersymmetry. We exhibit a new static 1/4 supersymmetric `kink-lump' solution
in which a string ends on a wall, and show that it has an effective realization
as a BIon of the D=4 super DBI-action. It is also shown to have a
time-dependent Q-kink-lump generalization which reduces to the Q-lump in a
limit corresponding to infinite BI magnetic field. All these 1/4 supersymmetric
sigma-model solitons are shown to be realized in M-theory as calibrated, or
`Q-calibrated', M5-branes in an M-monopole background.
