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The MIGHTEE survey utilizes the South African MeerKAT radio telescope to observe four extragalactic deep fields, with the aim of advancing our understanding of the formation and evolution of galaxies across cosmic time. MIGHTEE's frequency coverage encompasses the |$\textrm {H}\scriptstyle \mathrm{I}$|  line to a redshift of z |$\simeq$|  0.58, and OH megamasers to z |$\simeq$|  0.9. We present the MIGHTEE- |$\textrm {H}\scriptstyle \mathrm{I}$|  imaging products for the COSMOS field, using a total of 94.2 h on-target and a close-packed mosaic of 15 individual pointings. The spectral imaging covers two broad, relatively interference-free regions (960–1150 and 1290–1520 MHz) within MeerKAT's L -band, with up to 26 kHz spectral resolution (5.5 km s |$^{-1}$| at z  = 0). The median noise in the highest spectral resolution data is 74 μ Jy beam |$^{-1}$|⁠ , corresponding to a 5 |$\sigma$| |$\textrm {H}\scriptstyle \mathrm{I}$|  mass limit of 10 |$^{8.5}$|  M |$_{\odot }$| for a 300 km s |$^{-1}$| line at z  = 0.07. The mosaics cover |$\gt $| 4 deg |$^{2}$|⁠ , provided at multiple angular resolution / sensitivity pairings, with an angular resolution for |$\textrm {H}\scriptstyle \mathrm{I}$|  at z  = 0 of 12 arcsec. We describe the spectral line processing workflow that will be the basis for future MIGHTEE- |$\textrm {H}\scriptstyle \mathrm{I}$|  products, and validation of, and some early results from, the spectral imaging of the COSMOS field. We find no evidence for line emission at the position of the z  = 0.376 |$\textrm {H}\scriptstyle \mathrm{I}$|  line reported from the CHILES survey at a |$\gt $| 94 per cent confidence level, placing a 3 |$\sigma$| upper limit of 8.1 |$\times$|  10 |$^{9}$|  M |$_{\odot }$| on |$M_{\mathrm{HI}}$| for this galaxy. A public data release accompanies this article. [ABSTRACT FROM AUTHOR]