Narrow-line Seyfert 1 (NLSy1) galaxies are a peculiar class of type 1 active galactic nuclei (broad-line Seyfert 1 galaxies, hereinafter BLSy1). The X-ray properties of individual objects belonging to this class are often extreme and associated with accretion at high Eddington ratios. Here, we present a study on a sample of 14 NLSy1 galaxies selected at hard X-rays (>20 keV) from the fourth INTEGRAL/IBIS catalogue. The 20–100 keV IBIS spectra show hard-X-ray photon indices flatly distributed (Γ20−100 keV ranging from ∼1.3 to ∼3.6) with an average value of 〈Γ20−100 keV〉= 2.3 ± 0.7, compatible with a sample of hard-X-ray BLSy1 average slopes. Instead, NLSy1 galaxies show steeper spectral indices with respect to BLSy1 galaxies when broad-band spectra are considered. Indeed, we combine XMM–Newton and Swift/XRT with INTEGRAL/IBIS data sets to obtain a wide energy spectral coverage (0.3–100 keV). A constraint on the high energy cut-off and on the reflection component is achieved only in one source, SWIFT J2127.4+5654 (Ecut-off∼ 50 keV, R= 1.0+0.5− 0.4). Hard-X-ray-selected NLSy1 galaxies do not display particularly strong soft excess emission, while absorption fully or partially covering the continuum is often measured as well as Fe line emission features. Variability is a common trait in this sample, both at X-rays and at hard X-rays. The fraction of NLSy1 galaxies in the hard-X-ray sky is likely to be ∼15 per cent, in agreement with estimates derived in optically selected NLSy1 samples. We confirm the association of NLSy1 galaxies with small black hole masses with a peak at 107 M⊙ in the distribution; however, hard-X-ray NLSy1 galaxies seem to occupy the lower tail of the Eddington ratio distribution of classical NLSy1 galaxies.