Context. The dynamical structure of the atmosphere of Cepheids has been well studied in the optical. Several authors have found very interesting spectral features in the J band, but little data have been secured beyond 1.6 μm. However, such observations can probe different radial velocities and line asymmetry regimes, and are able to provide crucial insights into stellar physics. Aims. Our goal was to investigate the infrared line-forming region in the K-band domain, and its impact on the projection factor and the k-term of Cepheids. Methods. We secured CRIRES observations for the long-period Cepheid l Car, with a focus on the unblended spectral line NaI 2208.969 nm. We measured the corresponding radial velocities (by using the first moment method) and the line asymmetries (by using the bi-Gaussian method). These quantities are compared to the HARPS visible spectra we previously obtained on l Car. Results. The optical and infrared radial velocity curves show the same amplitude (only about 3% of difference), with a slight radial velocity shift of about 0.5 ± 0.3 km s⁻¹ between the two curves. Around the minimum radius (phase ≃ 0.9) the visible radial velocity curve is found in advance compared to the infrared one (phase lag), which is consistent with an infrared line forming higher in the atmosphere (compared to the visible line) and with a compression wave moving from the bottom to the top of the atmosphere during maximum outward velocity. The asymmetry of the K-band line is also found to be significantly different from that of the optical line.