The detection of five new fast radio bursts (FRBs) found in the 1.4-GHz High Time Resolution Universe high-latitude survey at Parkes, is presented. The rate implied is 7
(95 per cent) FRBs sky−1 d−1 above a fluence of 0.13 Jy ms for an FRB of 0.128 ms duration to 1.5 Jy ms for 16 ms duration. One of these FRBs has a two-component profile, in which each component is similar to the known population of single component FRBs and the two components are separated by 2.4 ± 0.4 ms. All the FRB components appear to be unresolved following deconvolution with a scattering tail and accounting for intrachannel smearing. The two-component burst, FRB 121002, also has the highest dispersion measure (1629 pc cm−3) of any FRB to-date. Many of the proposed models to explain FRBs use a single high-energy event involving compact objects (such as neutron-star mergers) and therefore cannot easily explain a two-component FRB. Models that are based on extreme versions of flaring, pulsing, or orbital events, however, could produce multiple component profiles. The compatibility of these models and the FRB rate implied by these detections is discussed.