We used K-band photometry, maximum rotational velocities derived from Fabry-Perot data and H i observed and predicted masses to study, for the first time, the K band, stellar and baryonic Tully-Fisher relations for galaxies in Hickson compact groups. We compared these relations with the ones defined for galaxies in less dense environments from the Gassendi HAlpha survey of Spirals and from a sample of gas-rich galaxies. We find that most of the Hickson compact group galaxies lie on the K-band Tully-Fisher relation defined by field galaxies with a few low-mass outliers, namely HCG 49b and HCG 96c, which appear to have had strong recent burst of star formation. The stellar Tully-Fisher relation for compact group galaxies presents a similar dispersion to that of the K-band relation, and it has no significant outliers when a proper computation of the stellar mass is done for the strongly star-forming galaxies. The scatter in these relations can be reduced if the gaseous component is taken into account, i.e. if a baryonic Tully-Fisher relation is considered. In order to explain the positions of the galaxies off the K-band Tully-Fisher relation, we favour a scenario in which their luminosities are brightened due to strong star formation or AGN activity. We argue that strong bursts of star formation can affect the B- and K-band luminosities of HCG 49b and HCG 96c and in the case of the latter also AGN activity may affect the K-band magnitude considerably, without affecting their total masses.