We present a new investigation of the intergalactic medium near reionization using dark gaps in the Ly{beta} forest. With its lower optical depth, Ly{beta} offers a potentially more sensitive probe to any remaining neutral gas compared to the commonly used Ly{alpha} line. We identify dark gaps in the Ly{beta} forest using spectra of 42 QSOs at z_em_>5.5, including new data from the XQR-30 VLT Large Programme. Approximately 40% of these QSO spectra exhibit dark gaps longer than 10h^-1^Mpc at z~5.8. By comparing the results to predictions from simulations, we find that the data are broadly consistent both with models where fluctuations in the Ly{alpha} forest are caused solely by ionizing ultraviolet background fluctuations and with models that include large neutral hydrogen patches at z<6 due to a late end to reionization. Of particular interest is a very long (L=28h^-1^Mpc) and dark ({tau}eff>~6) gap persisting down to z~5.5 in the Ly{beta} forest of the z=5.85 QSO PSO J025-11. This gap may support late reionization models with a volume-weighted average neutral hydrogen fraction of <xHI>>~5% by z=5.6. Finally, we infer constraints on <xHI> over 5.5<~z<~6.0 based on the observed Ly{beta} dark gap length distribution and a conservative relationship between gap length and neutral fraction derived from simulations. We find <xHI><=0.05, 0.17, and 0.29 at z~5.55, 5.75, and 5.95, respectively. These constraints are consistent with models where reionization ends significantly later than z=6.