This paper presents a spectral survey of diffuse interstellar bands (DIBs) in the NIR range, from 0.9 to 2.5 micron. The observations were designed to detect new DIBs, confirm previously proposed NIR DIBs, and characterise their behaviour with respect to known line-of-sight properties (including the optical DIBs present in our spectra). X-shooter at the VLT was used to obtained medium-resolution spectra of eight known DIB targets and one telluric reference star. In addition to the known 9577, 9632, 10780, 11797, and 13175 Angstroms NIR DIBs, we confirm 9 out of the 13 NIR DIBs that were presented by Geballe and co-workers in 2011. Furthermore, we report 12 new NIR DIB candidates. The strengths of the strongest NIR DIBs show a general correlation with reddening, E(B-V), but with a large scatter. Several NIR DIBs are more strongly correlated with the 5780 Angstroms DIB strength than with E(B-V); this is especially the case for the 15268 Angstroms DIB. The NIR DIBs are strong: the summed equivalent widths of the five strongest NIR DIBs represent a few percent of the total equivalent width of the entire average DIB spectrum (per unit reddening). The NIR DIBs towards the translucent cloud HD 147889 are all weak with respect to the general trend. No direct match was found between observed NIR DIBs and laboratory matrix-isolation spectroscopic data of PAHs. The strong correlation between the 5780-15268 DIB pair implies that (Nf)_5780 / (Nf)_15268 = 14. However, the reduced strength of the 15268 Angstroms DIB in HD 147889 rules out a common carrier for these two DIBs. Since the ionisation fraction for small PAHs in this translucent cloud is known to be low compared to diffuse clouds, the weakness of the 15268 Angstrom DIB suggests that an ionised species could be the carrier of this NIR DIB. (abridged).

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