  We present a new investigation of the intergalactic medium (IGM) near
reionization using dark gaps in the Lyman-$\beta$ (Ly$\beta$) forest. With its
lower optical depth, Ly$\beta$ offers a potentially more sensitive probe to any
remaining neutral gas compared to commonly used Ly$\alpha$ line. We identify
dark gaps in the Ly$\beta$ forest using spectra of 42 QSOs at $z_{\rm em}>5.5$,
including new data from the XQR-30 VLT Large Programme. Approximately $40\%$ of
these QSO spectra exhibit dark gaps longer than $10h^{-1}{\rm Mpc}$ at
$z\simeq5.8$. 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 (UVB)
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
($L=28h^{-1}{\rm Mpc}$) and dark ($\tau_{\rm eff} \gtrsim 6$) gap persisting
down to $z\simeq 5.5$ in the Ly$\beta$ forest of the $z_{\rm}=5.85$ QSO PSO
J025$-$11. This gap may support late reionization models with a volume-weighted
average neutral hydrogen fraction of $ \langle x_{\rm HI}\rangle \gtrsim 5\%$
by $z=5.6$. Finally, we infer constraints on $\langle x_{\rm HI}\rangle$ over
$5.5 \lesssim z \lesssim 6.0$ 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 $\langle x_{\rm HI}\rangle \leq
0.05$, 0.17, and 0.29 at $z\simeq 5.55$, 5.75, and 5.95, respectively. These
constraints are consistent with models where reionization ends significantly
later than $z = 6$.
