  In the Einstein-bumblebee gravity, the Lorentz symmetry is spontaneously
broken by a vector field. In this paper, we attempt to test the Lorentz
symmetry via the observation of the shadow cast by the Kerr-like black hole
with or without plasma. A novel phenomenon of the Lorentz-violating parameter
on the shadow is observed. The result shows that when the observer gradually
moves from the poles to the equatorial plane, the shadow radius $R_{\rm s}$
firstly decreases and then increases with the Lorentz-violating parameter. Such
nonmonotonic behavior provides us an important understanding on the black hole
shadow in the Einstein-bumblebee gravity. Besides, three more distortion
observables are calculated, and found to increase with the Lorentz-violating
parameter. Moreover, when a homogeneous plasma is present, the motion of the
photon is analyzed. We further observe that the refractive index shrinks the
size, while enhances the deformation of the shadow. Finally, adopting the
observed data of the diameter of M87$^*$, we find the refractive index is more
favored in (0.914, 1).
