  We report on systematic temperature- and magnetic field-dependent studies of
the EuGa$_4$ binary compound, which crystallizes in a centrosymmetric
tetragonal BaAl$_4$-type structure with space group $I4/mmm$. The electronic
properties of EuGa$_4$ single crystals, with an antiferromagnetic (AFM)
transition at $T_\mathrm{N} \sim 16.4$ K, were characterized via electrical
resistivity and magnetization measurements. A giant nonsaturating
magnetoresistance was observed at low temperatures, reaching $\sim 7 \times
10^4$ % at 2 K in a magnetic field of 9 T. In the AFM state, EuGa$_4$ undergoes
a series of metamagnetic transitions in an applied magnetic field, clearly
manifested in its field-dependent electrical resistivity. Below $T_\mathrm{N}$,
in the $\sim$4-7 T field range, we observe also a clear hump-like anomaly in
the Hall resistivity which is part of the anomalous Hall resistivity. We
attribute such a hump-like feature to the topological Hall effect, usually
occurring in noncentrosymmetric materials known to host topological spin
textures (as e.g., magnetic skyrmions). Therefore, the family of materials with
a tetragonal BaAl$_4$-type structure, to which EuGa$_4$ and EuAl$_4$ belong,
seems to comprise suitable candidates on which one can study the interplay
among correlated-electron phenomena (such as charge-density wave or exotic
magnetism) with topological spin textures and topologically nontrivial bands.
