Context. Double-peaked emission lines are observed in a small percentage of active galactic nuclei (AGN). These lines allow the determination of fundamental properties of the line-emitting region, known as the broad-line region (BLR). Aims. We investigated the structure and kinematics of the BLR in the nearby Seyfert galaxy NGC 4593 through an analysis of the near-infrared (NIR) line blend of Ca II lambda 8498, lambda 8542, lambda 8662, and O I lambda 8446 observed in a 2019 VLT/MUSE spectrum. Methods. We performed a detailed decomposition of the NIR Ca II triplet and O I lambda 8446 blend, extracting clean profiles of Ca II lambda 8498, lambda 8542, lambda 8662, and O I lambda 8446. We then fitted Ca II lambda 8662 with a relativistic elliptical line-emitting accretion disk model. Results. The extracted line profiles are double-peaked with a full width at half maximum (FWHM) of similar to 3700 km s(-1) and exhibit a redward asymmetry with a red-to-blue peak ratio of 4:3. The Ca II triplet lines have an intensity ratio of 1:1:1 and show no evidence of a central narrow or intermediate-width component. The profiles of Ca II and O I are remarkably similar, suggesting a common region of origin. Given the 1:1:1 ratio of the Ca II triplet, this region is likely a high-density emission zone, and the Ca II lambda 8662 profile is well described by a mildly eccentric, low-inclination relativistic line-emitting disk with minimal internal turbulence. The profile represents one of the clearest kinematic signatures of a relativistic disk observed in BLR emission lines to date. Conclusions. The double-peaked profiles of the NIR Ca II triplet and O I lambda 8446 in NGC 4593 represent the first detection of double-peaked Ca II and O I lambda 8446 in a nontransient AGN spectrum. The minimal intrinsic turbulence (the lowest value reported for an AGN emission line to date) and the absence of narrow or intermediate-width components in Ca II lambda 8662 make it a powerful diagnostic tool of BLR structure and kinematics. Further investigations of the profiles of Ca II and O I in other AGN are recommended to better constrain BLR properties and the nature of the underlying accretion flow.