The SPIRou Legacy Survey. Detection of a nearby world orbiting in the habitable zone of Gl 725 B achieved by correcting strong telluric contamination in near-infrared radial velocities with wapiti

M dwarfs are prime targets in the search for exoplanets because of their prevalence and because low-mass planets can be better detected with radial velocity (RV) methods. In particular, the near-infrared (NIR) spectral domain offers an increased RV sensitivity and potentially reduced stellar activity signals. Precise NIR RV measurements are strongly affected by telluric absorption lines from the Earth's atmosphere, however. We searched for planets orbiting Gl,725,B, a nearby late-M dwarf located at 3.5,pc, using high-precision SPIRou RV observations. We also assessed the effect of telluric contamination on these measurements and evaluated the performance of the weighted principal component analysis reconstruction ( wapiti ) method, which is a weighted principal component analysis (wPCA) approach designed to mitigate these systematics and to improve the sensitivity of planet detections. Using synthetic and observational SPIRou data, we simulated the effect of telluric lines on RV data under varying barycentric Earth radial velocity (BERV) conditions. We then applied the wapiti method for identifying and correcting telluric-induced systematics in line-by-line RVs. The method was tested through an injection-recovery test on simulated data and was subsequently applied to real SPIRou observations of Gl,725,B. wapiti successfully corrects telluric contamination in simulated and real datasets. This enhances the detectability and accuracy of planetary signals. In the corrected Gl,725,B dataset, we identified a two-planet system composed of a candidate inner planet (Gl 725 Bb), with periods of $4.765 ± 0.004$ days and a semi-amplitude of $1.4 ± 0.3$,m⋅s^-1, and a confirmed planet Gl,725,Bc, with a period of $37.90 ± 0.17$ days and a semi-amplitude of $1.7 ± 0.3$,m⋅s^-1. Their minimum mass is $1.5 ± 0.4$,M_⊕ and of $3.5 ± 0.7$,M_⊕, respectively, and the outer planet is located in the habitable zone of its host star. Using a multi-dimensional Gaussian process framework to model and correct for stellar activity, we also recovered a stellar rotation period of $105.1 ± 3.3$ days.