  We investigate static and spherically symmetric stars disformally coupled to
a scalar field. The scalar field is assumed to be shift symmetric, and hence
the conformal and disformal factors of the metric coupled to matter are
dependent only on the kinetic term of the scalar field. Assuming that the
scalar field is linearly dependent on time, we consider a general
shift-symmetric scalar-tensor theory and a general form of the matter
energy-momentum tensor that allows for the anisotropic pressure and the heat
flux in the radial direction. This is a natural starting point in light of how
the gravitational field equations and the energy-momentum tensor transform
under a disformal transformation. By inspecting the structure of the
hydrostatic equilibrium equation in the presence of the derivative-dependent
conformal and disformal factors, we show that the energy density and tangential
pressure must vanish at the surface of a star. This fact is used to prove the
disformal invariance of the surface of a star, which was previously subtle and
unclear. We then focus on the shift-symmetric k-essence disformally coupled to
matter, and study the interior and exterior metric functions and scalar-field
profile in more detail. It is found that there are two branches of the solution
depending on the velocity of the scalar field. The disformally-related metric
functions of the exterior spacetime are also discussed.
