English: Hyperfine-structure parameters and isotope shifts for the 795-nm atomic transitions in 217,218,219At have been measured at CERN-ISOLDE, using the in-source resonance-ionization spectroscopy technique. Magnetic dipole and electric quadrupole moments, and changes in the nuclear mean-square charge radii, have been deduced. A large inverse odd-even staggering in radii, which may be associated with the presence of octupole collectivity, has been observed. Namely, the radius of the odd-odd isotope 218At has been found to be larger than the average of its even-N neighbors, 217,219At. The discrepancy between the additivity-rule prediction and experimental data for the magnetic moment of 218At also supports the possible presence of octupole collectivity in the considered nuclei.