K-dwarf stars are promising targets in the exploration of potentially habitable planets. Their properties, falling between G and M dwarfs, provide an optimal trade off between habitability prospect and ease of detection. The KOBE experiment is a blind-search survey exploiting this niche, monitoring the radial velocity of 50 late-type K-dwarf stars using the CARMENES spectrograph with an observational strategy designed to detect planets in the habitable zone of their system. In this paper, we exploited the KOBE data set to characterize planetary signals in the K7 V star HIP 5957 (KOBE-1) and to constrain the planetary population within its habitable zone. We used 82 CARMENES spectra with a time span of three years. We employed the generalized Lomb-Scargle periodogram to search for significant periodic signals compatible with Keplerian motion on KOBE-1. We conducted model comparison in a Bayesian framework to ensure the significance of the planetary model over alternative configurations of lower complexity. We also inspected two available TESS sectors in search of planetary signals. We identified two signals at Pb=8.5d and Pc=29.7d. We confirmed their planetary nature through ruling out other non-planetary configurations. Their minimum masses are 8.80+/-0.76M_{Earth} (KOBE-1 b), and 12.4+/-1.1M{Earth} (KOBE-1 c), corresponding to absolute masses within the planetary regime to a high certainty (>99.7%). By analyzing the sensitivity of the CARMENES time series to additional signals, we discarded planets above 8.5M{Earth}_ within the habitable zone. We identified a single transit-like feature in TESS whose origin is still uncertain, but it is compatible within 1-sigma with a transit from planet c. The KOBE-1 multi-planetary system, consisting of a relatively quiet K7-dwarf hosting two sub-Neptune-minimum-mass planets, sets the first discovery of the KOBE experiment. We explored future prospects for characterizing this system, concluding that Gaia DR4 will be insensitive to their astrometric signature, while nulling interferometry with LIFE could be capable of directly imaging both planets and characterize their atmospheres.