  We explore the notion of isometries in non-Riemannian geometries. Such
geometries include and generalise the backgrounds of non-relativistic string
theory, and they can be naturally described using the formalism of double field
theory. Adopting this approach, we first solve the corresponding Killing
equations for constant flat non-Riemannian backgrounds and show that they admit
an infinite-dimensional algebra of isometries which includes a particular type
of supertranslations. These symmetries correspond to known worldsheet Noether
symmetries of the Gomis-Ooguri non-relativistic string, which we now interpret
as isometries of its non-Riemannian doubled background. We further consider the
extension to supersymmetric double field theory and show that the corresponding
Killing spinors can depend arbitrarily on the non-Riemannian directions,
leading to "supersupersymmetries" that square to supertranslations.
