  Extended metric-Palatini gravity, quadratic in the antisymmetric part of the
affine curvature, is known to lead to the general relativity plus a geometric
Proca field. The geometric Proca, equivalent of the non-metricity vector in the
torsion-free affine connection, qualifies to be a distinctive signature of the
affine curvature. In the present work, we explore how shadow and photon motion
near black holes can be used to probe the geometric Proca field. To this end,
we derive static spherically symmetric field equations of this
Einstein-geometric Proca theory, and show that it admits black hole solutions
in asymptotically AdS background. We perform a detailed study of the optical
properties and shadow of this black hole and contrast them with the
observational data by considering black hole environments with and without
plasma. As a useful astrophysical application, we discuss constraints on the
Proca field parameters using the observed angular size of the shadow of
supermassive black holes M87$^*$ and Sgr A$^*$ in both vacuum and plasma cases.
Overall, we find that the geometric Proca can be probed via the black hole
observations.
