We present the selection of the Jodrell Bank Flat-spectrum (JBF) radio source sample, which is designed to reduce the uncertainties in the Cosmic Lens All-Sky Survey (CLASS) gravitational lensing statistics arising from the lack of knowledge about the parent population luminosity function. From observations at 4.86 GHz with the Very Large Array, we have selected a sample of 117 flat-spectrum radio sources with flux densities greater than 5 mJy. These sources were selected in a similar manner to the CLASS complete sample and are therefore representative of the parent population at low flux densities. The vast majority (~90 per cent) of the JBF sample are found to be compact on the arcsec scales probed here and show little evidence of any extended radio jet emission. Using the JBF and CLASS complete samples, we find the differential number counts slope of the parent population above and below the CLASS 30-mJy flux density limit to be -2.07 +/- 0.02 and -1.96 +/- 0.12, respectively.