We present a new sample of galaxy-scale strong gravitational-lens candidates, selected from 904 square degrees of Data Release 4 of the Kilo-Degree Survey (KiDS), i.e., the Lenses in the Kilo-Degree Survey (LinKS) sample. We apply two Convolutional Neural Networks (ConvNets) to $sim88,000$ colour-magnitude selected luminous red galaxies yielding a list of 3500 strong-lens candidates. This list is further down-selected via human inspection. The resulting LinKS sample is composed of 1983 rank-ordered targets classified as potential lens candidates by at least one inspector. Of these, a high-grade subsample of 89 targets is identified with potential strong lenses by all inspectors. Additionally, we present a collection of another 200 strong lens candidates discovered serendipitously from various previous ConvNet runs. A straightforward application of our procedure to future Euclid or LSST data can select a sample of $sim3000$ lens candidates with less than 10 per cent expected false positives and requiring minimal human intervention.