We present a new investigation of the intergalactic medium (IGM) near the end of reionization using "dark gaps" in the Lyman-alpha (Ly$\alpha$) forest. Using spectra of 55 QSOs at $zem>5.5$, including new data from the XQR-30 VLT Large Programme, we identify gaps in the Ly$\alpha$ forest where the transmission averaged over 1 comoving $h-1Mpc$ bins falls below 5%. Nine ultra-long ($L>80 h-1Mpc$) dark gaps are identified at . In addition, we quantify the fraction of QSO spectra exhibiting gaps longer than $30 h-1Mpc$, $F30$, as a function of redshift. We measure $F30\simeq 0.9$, 0.6, and 0.15 at $z=6.0$, 5.8, and 5.6, respectively, with the last of these long dark gaps persisting down to $z\simeq 5.3$. Comparing our results with predictions from hydrodynamical simulations, we find that the data are consistent with models wherein reionization extends significantly below redshift six. Models wherein the IGM is essentially fully reionized that retain large-scale fluctuations in the ionizing UV background at $z\lesssim 6$ are also potentially consistent with the data. Overall, our results suggest that signature of reionization in the form of islands of neutral hydrogen and/or large-scale fluctuations in the ionizing background remain present in the IGM until at least $z\simeq 5.3$.