Inspiraling and merging binary neutron stars (BNSs) are important sources of both gravitational waves and coincident electromagnetic counterparts. If the BNS total mass is larger than a threshold value, a black hole ensues promptly after merger. Through a statistical study in conjunction with recent LIGO/Virgo constraints on the nuclear equation of state, we estimate that up to ∼25% of BNS mergers may result in prompt collapse. Moreover, we find that most models of the BNS mass function we study here predict that the majority of prompt-collapse BNS mergers have q≳0.8. Prompt-collapse BNS mergers with mass ratio q≳0.8 may not be accompanied by detectable kilonovae or short gamma-ray bursts, because they unbind a negligible amount of mass and form negligibly small accretion disks onto the remnant black hole. We call such BNS mergers 'orphan.' However, recent studies have found that 1041-43(Bp/1012  G)2  erg s−1 electromagnetic signals can be powered by magnetospheric interactions several milliseconds prior to merger. Moreover, the energy stored in the magnetosphere of an orphan BNS merger remnant will be radiated away in O(1  ms). Through simulations in full general relativity of BNSs endowed with an initial dipole magnetosphere, we find that the energy in the magnetosphere following black hole formation is EB∼1039-41(Bp/1012  G)2  erg. Radiating ∼1% of EB in 1 ms, as has been found in previous studies, matches the premerger magnetospheric luminosity. These magnetospheric signals are not beamed, and their duration and power agrees with those of nonrepeating fast radio bursts (FRBs). These results combined with our statistical study suggest that a nonrepeating FRB may be the most likely electromagnetic counterpart of prompt-collapse BNSs. Detection of a nonrepeating FRB coincident with gravitational waves from a BNS merger could settle the extragalactic origin of a fraction FRBs and could be used to place constraints on the nuclear equation of state. FRBs can also initiate triggered searches for weak signals in the LIGO/Virgo data.