  We analyze JWST/NIRSpec observations of the CO rovibrational v=1-0 band at
4.67um around the dust-embedded southern active galactic nucleus (AGN) of
NGC3256 (d=40Mpc; L(IR)=10^11.6 Lsun). We classify the CO v=1-0 spectra into
three categories based on the behavior of P- and R-branches of the band: (a)
both branches in absorption toward the nucleus; (b) P-R asymmetry (P-branch in
emission and R-branch in absorption) along the disk of the galaxy; and (c) both
branches in emission in the outflow region above and below the disk. In this
paper, we focus on the outflow. The CO v=1-0 emission can be explained by the
vibrational excitation of CO in the molecular outflow by the bright mid-IR
~4.7um continuum from the AGN up to r~250pc. We model the ratios between the
P(J+2) and R(J) transitions of the band to derive the physical properties
(column density, kinetic temperature, and CO-to-H2 conversion factor, alpha_CO)
of the outflowing gas. We find that the 12CO v=1-0 emission is optically thick
for J<4, while the 13CO v=1-0 emission remains optically thin. From the
P(2)/R(0) ratio, we identify a temperature gradient in the outflow from >40K in
the central 100pc to <15K at 250pc sampling the cooling of the molecular gas in
the outflow. We used three methods to derive alpha_CO in eight 100pc (0.5")
apertures in the outflow by fitting the P(J+2)/R(J) ratios with non-LTE models.
We obtain low median alpha_CO factors (0.40-0.61) x 3.2e-4/[CO/H2] Msun (K
km/s/pc2)^-1 in the outflow regions. This implies that outflow rates and
energetics might be overestimated if a ULIRG-like alpha_CO, which is 1.3-2
times larger, is assumed. The reduced alpha_CO can be explained if the
outflowing molecular clouds are not virialized. We also report the first
extragalactic detection of a broad (sigma=0.0091um) spectral feature at 4.645um
associated with aliphatic deuterium on polycyclic aromatic hydrocarbons
(D_n-PAH).
