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The GOES X1 flare SOL2014-10-25T17:08:00 was a three-ribbon solar flare observed with IRIS in the near and far ultraviolet. One of the flare ribbons crossed a sunspot umbra, producing a dramatic, $\sim1000$% increase in the near-ultraviolet (NUV) continuum radiation. We comprehensively analyze the ultraviolet spectral data of the umbral flare brightenings, which provide new challenges for radiative-hydrodynamic modeling of the chromospheric velocity field and the white-light continuum radiation. The emission line profiles in the umbral flare brightenings exhibit redshifts and profile asymmetries, but these are significantly smaller than in another, well-studied X-class solar flare. We present a ratio of the NUV continuum intensity to the Fe II 2814.45 Ang intensity. This continuum-to-line ratio is a new spectral diagnostic of significant heating at high column mass (log $m/$[g cm$^{-2}] >-2$) during solar flares because the continuum and emission line radiation originate from relatively similar temperatures but moderately different optical depths. The full spectral readout of these IRIS data also allow for a comprehensive survey of the flaring NUV landscape: in addition to many lines of Fe II and Cr II, we identify a new solar flare emission line, He I $\lambda2829.91$ (as previously identified in laboratory and early-type stellar spectra). The Fermi/GBM hard X-ray data provide inputs to radiative-hydrodynamic models (which will be presented in Paper II) in order to better understand the large continuum-to-line ratios, the origin of the white-light continuum radiation, and the role of electron beam heating in the low atmosphere., Comment: 19 pg, 10 fig, 2 appendices. Accepted for publication in ApJ