  Semichiral sigma models with a four-dimensional target space do not support
extended N=(4,4) supersymmetries off-shell arXiv:0903.2376, arXiv:0912.4724. We
contribute towards the understanding of the non-manifest on-shell
transformations in (2,2) superspace by analyzing the extended on-shell
supersymmetry of such models and find that a rather general ansatz for the
additional supersymmetry (not involving central charge transformations) leads
to hyperk\"ahler geometry. We give non-trivial examples of these models.
