Narrow-line Seyfert 1 (NLS1) galaxies are believed to be active galactic nuclei (AGN) in the early stages of their evolution. Some dozens of them have been found to host relativistic jets, whilst the majority has not even been detected in radio, emphasising the heterogeneity of the class in this band. In this paper, our aim is to determine the predominant source of radio emission in a sample of 44 NLS1s, selected based on their extended kpc-scale radio morphologies at 5.2 GHz. We accomplish this by analysing their spatially resolved radio spectral index maps, centred at 5.2 GHz. In addition, we utilise several diagnostics based on mid-infrared emission to estimate the star formation activity of their host galaxies. These data are complemented by archival data to draw a more complete picture of each source. We find an extraordinary diversity among our sample. Approximately equal fractions of our sources can be identified as AGN-dominated, composite, and host-dominated. Among the AGN-dominated sources are a few NLS1s with very extended jets, reaching distances of tens of kpc from the nucleus. One of these, J0814+5609, hosts the most extended jets found in an NLS1 so far. We also identify five NLS1s that could be classified as compact steep-spectrum sources. We further conclude that due to the variety seen in NLS1s simple proxies, such as the star formation diagnostics also employed in this paper, and the radio loudness parameter, are not ideal tools for characterising NLS1s. We emphasise the necessity of examining NLS1s as individuals, instead of making assumptions based on their classification. When these issues are properly taken into account, NLS1s offer an exceptional environment to study the interplay of the host galaxy and several AGN-related phenomena, such as jets and outflows. [Abstract abridged.]
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