We use Gaia EDR3 data to identify stars associated with six classical dwarf spheroidals (dSphs) (Draco, Ursa Minor, Sextans, Sculptor, Fornax, Carina) at their outermost radii, beyond their nominal King stellar limiting radius. For all of the dSphs examined, we find radial velocity matches with stars residing beyond the King limiting radius and with |${\gt}50{{\ \rm per\ cent}}$| astrometric probability (four in Draco, two in Ursa Minor, eight in Sextans, two in Sculptor, 12 in Fornax, and five in Carina), indicating that these stars are associated with their respective dSphs at high probability. We compare the positions of our candidate 'extra-tidal' stars with the orbital tracks of the galaxies, and identify stars, both with and without radial velocity matches, that are consistent with lying along the orbital track of the satellites. However, given the small number of candidate stars, we cannot make any conclusive statements about the significance of these spatially correlated stars. Cross matching with publicly available catalogues of RR Lyrae, we find one RR Lyrae candidate with |${\gt}50{{\ \rm per\ cent}}$| astrometric probability outside the limiting radius in each of Sculptor and Fornax, two such candidates in Draco, nine in Ursa Minor, seven in Sextans, and zero in Carina. Follow-up spectra on all of our candidates, including possible metallicity information, will help confirm association with their respective dSphs, and could represent evidence for extended stellar haloes or tidal debris around these classical dSphs. [ABSTRACT FROM AUTHOR]