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(Abridged) Type I superluminous supernovae (SLSNe I) are rare, powerful explosions whose mechanism and progenitors remain elusive. SLSNe I show a preference for low-metallicity, actively star-forming dwarf galaxies. We investigate whether the hosts of SLSNe I show increased evidence for interaction. We use a sample of 42 SLSN I images obtained with $\textit{HST}$ and measure the number of companion galaxies by counting the objects detected within a given radius from the host. As a comparison, we used two Monte Carlo-based methods to estimate the expected average number of companion objects in the same images, as well as a sample of 32 galaxies that have hosted long gamma-ray bursts (GRBs). About 50% of SLSN I hosts have at least one major companion (within a flux ratio of 1:4) within 5 kpc. The average number of major companions per SLSN I host galaxy is $0.70^{+0.19}_{-0.14}$. Our Monte Carlo comparison methods yield a lower number of companions for random objects of similar brightness in the same image or for the SLSN host after randomly redistributing the sources in the same image. The Anderson-Darling test shows that this difference is statistically significant independent of the redshift range. The same is true for the projected distance distribution of the companions. The SLSN I hosts are, thus, found in areas of their images, where the object number density is greater than average. SLSN I hosts have more companions than GRB hosts ($0.44^{+0.25}_{-0.13}$ companions per host distributed over 25% of the hosts) but the difference is not statistically significant. The difference between their separations is, however, marginally significant. The dwarf galaxies hosting SLSNe I are often part of interacting systems. This suggests that SLSNe I progenitors are formed after a recent burst of star formation. Low metallicity alone cannot explain this tendency.