  For a theory in which a scalar field $\phi$ has a nonminimal derivative
coupling to the Einstein tensor $G_{\mu \nu}$ of the form $\phi\,G_{\mu
\nu}\nabla^{\mu}\nabla^{\nu} \phi$, it is known that there exists a branch of
static and spherically-symmetric relativistic stars endowed with a scalar hair
in their interiors. We study the stability of such hairy solutions with a
radial field dependence $\phi(r)$ against odd- and even-parity perturbations.
We show that, for the star compactness ${\cal C}$ smaller than $1/3$, they are
prone to Laplacian instabilities of the even-parity perturbation associated
with the scalar-field propagation along an angular direction. Even for ${\cal
C}>1/3$, the hairy star solutions are subject to ghost instabilities. We also
find that even the other branch with a vanishing background field derivative is
unstable for a positive perfect-fluid pressure, due to nonstandard propagation
of the field perturbation $\delta \phi$ inside the star. Thus, there are no
stable star configurations in derivative coupling theory without a standard
kinetic term, including both relativistic and nonrelativistic compact objects.
