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We present 637 low-redshift optical spectra collected by the Berkeley Supernova Ia Program (BSNIP) between 2009 and 2018, almost entirely with the Kast double spectrograph on the Shane 3~m telescope at Lick Observatory. We describe our automated spectral classification scheme and arrive at a final set of 626 spectra (of 242 objects) that are unambiguously classified as belonging to Type Ia supernovae (SNe~Ia). Of these, 70 spectra of 30 objects are classified as spectroscopically peculiar (i.e., not matching the spectral signatures of "normal" SNe~Ia) and 79 SNe~Ia (covered by 328 spectra) have complementary photometric coverage. The median SN in our final set has one epoch of spectroscopy, has a redshift of 0.0208 (with a low of 0.0007 and high of 0.1921), and is first observed spectroscopically 1.1 days after maximum light. The constituent spectra are of high quality, with a median signal-to-noise ratio of 31.8 pixel$^{-1}$, and have broad wavelength coverage, with $\sim 95\%$ covering at least 3700--9800~Å. We analyze our dataset, focusing on quantitative measurements (e.g., velocities, pseudo-equivalent widths) of the evolution of prominent spectral features in the available early-time and late-time spectra. The data are available to the community, and we encourage future studies to incorporate our spectra in their analyses.