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We present the largest and most homogeneous collection of near-infrared (NIR) spectra of Type Ia supernovae (SNe Ia): 339 spectra of 98 individual SNe obtained as part of the Carnegie Supernova Project-II. These spectra, obtained with the FIRE spectrograph on the 6.5 m Magellan Baade telescope, have a spectral range of 0.8--2.5 $μ$m. Using this sample, we explore the NIR spectral diversity of SNe Ia and construct a template of spectral time series as a function of the light-curve-shape parameter, color stretch $s_{BV}$. Principal component analysis is applied to characterize the diversity of the spectral features and reduce data dimensionality to a smaller subspace. Gaussian process regression is then used to model the subspace dependence on phase and light-curve shape and the associated uncertainty. Our template is able to predict spectral variations that are correlated with $s_{BV}$, such as the hallmark NIR features: Mg II at early times and the $H$-band break after peak. Using this template reduces the systematic uncertainties in K-corrections by ~90% compared to those from the Hsiao template. These uncertainties, defined as the mean K-correction differences computed with the color-matched template and observed spectra, are on the level of $4\times10^{-4}$ mag on average. This template can serve as the baseline spectral energy distribution for light-curve fitters and can identify peculiar spectral features that might point to compelling physics. The results presented here will substantially improve future SN~Ia cosmological experiments, for both nearby and distant samples.