The motion of doped electrons or holes in an antiferromagnetic lattice with strong on-site Coulomb interactions touches one of the most fundamental open problems in contemporary condensed matter physics. The doped charge may strongly couple to elementary spin excitations, resulting in a dressed quasiparticle which is subject to confinement. This "spin polaron" possesses internal degrees of freedom with a characteristic "ladder" excitation spectrum. Despite its fundamental importance for understanding higherature superconductivity, clear experimental spectroscopic signatures of these internal degrees of freedom are scarce. Here, we present scanning tunneling spectroscopy results of the spin-orbit-induced Mott insulator Sr₂IrO₄. Our spectroscopy data reveal distinct shoulder-like features for occupied and unoccupied states beyond a measured Mott gap of Δ≈620 meV. Using the self-consistent Born approximation we assign the anomalies in the unoccupied states to the spin-polaron ladder spectrum with excellent quantitative agreement and estimate the Coulomb repulsion U=2.05...2.28 eV in this material. These results confirm the strongly correlated electronic structure of this compound and underpin the previously conjectured paradigm of emergent unconventional superconductivity in doped Sr₂IrO₄.