The coupling between bound quantum states and those in the continuum is of high theoretical interest. Experimental studies of bound drip-line nuclei provide ideal testing grounds for such investigations since they, due to the feeble binding energy of their valence particles, are easy to excite into the continuum. In this Letter, continuum states in the heaviest particle-stable Be isotope, 14Be, are studied by employing the method of inelastic proton scattering in inverse kinematics. New continuum states are found at excitation energies E*=3.54(16) MeV and E*=5.25(19) MeV. The structure of the earlier known 2(1)+ state at 1.54(13) MeV was confirmed with a predominantly (0d5/2)2 configuration while there is very clear evidence that the 2(2)+ state has a predominant (1s1/2, 0d5/2) structure with a preferential three-body decay mechanism. The region at about 7 MeV excitation shows distinct features of sequential neutron decay via intermediate states in 13Be. This demonstrates that the increasing availability of energetic beams of exotic nuclei opens up new vistas for experiments leading towards a new understanding of the interplay between bound and continuum states.