We investigate the morphology of the [CII] emission in a sample of “normal” star-forming galaxies at 5 < z < 7.2 in relation to their UV (rest-frame) counterpart. We use new ALMA observations of galaxies at z ∼ 6 − 7, as well as a careful re-analysis of archival ALMA data. In total 29 galaxies were analysed, 21 of which are detected in [CII]. For several of the latter the [CII] emission breaks into multiple components. Only a fraction of these [CII] components, if any, is associated with the primary UV systems, while the bulk of the [CII] emission is associated either with fainter UV components, or not associated with any UV counterpart at the current limits. By taking into account the presence of all these components, we find that the L[CII]-SFR relation at early epochs is fully consistent with the local relation, but it has a dispersion of 0.48±0.07 dex, which is about two times larger than observed locally. We also find that the deviation from the local L[CII]-SFR relation has a weak anti-correlation with the EW(Lyα). The morphological analysis also reveals that [CII] emission is generally much more extended than the UV emission. As a consequence, these primordial galaxies are characterised by a [CII] surface brightness generally much lower than expected from the local Σ[CII]−ΣSFR relation. These properties are likely a consequence of a combination of different effects, namely: gas metallicity, [CII] emission from obscured star-forming regions, strong variations of the ionisation parameter, and circumgalactic gas in accretion or ejected by these primeval galaxies.