We still do not know what causes aspherical planetary nebula (PN) morphologies. A plausible hypothesis is that they are due to the presence of a close stellar or substellar companion. So far, only ~40 binary central stars of PN have been detected, almost all of them with such short periods that their binarity is revealed by photometric variability. Here we have endeavoured to discover binary central stars at any separation, thus determining the unbiased binary fraction of central stars of PN. This number, when compared to the binary fraction of the presumed parent population, can give a first handle on the origin of PN. By detecting the central stars in the I band we have searched for cool companions. We have found that 30 per cent of our sample have an I-band excess detected between 1 and a few σ, possibly denoting companions brighter than M3-4V and with separations smaller than ~1000 au. By accounting for the undetectable companions, we determine a debiased binary fraction of 67-78 per cent for all companions at all separations. We compare this number to a main-sequence binary fraction of (50 ± 4) per cent determined for spectral types F6V-G2V, appropriate if the progenitors of today's PN central star population are indeed the F6V-G2V stars. The error on our estimate cannot be constrained tightly, but we determine it to be between 10 and 30 per cent. We conclude that the central star binary fraction may be larger than expected from the putative parent population. However, this result is based on a sample of 27 bona fide central stars and should be considered preliminary. The success of the I-band method rests critically on high-precision photometry and a reasonably large sample. From a similar analysis, using the more sensitive J band of a subset of 11 central stars, the binary fraction is 54 per cent for companions brighter than ~M5-6V and with separations smaller than about 900 au. Debiasing this number in the same way as was done for the I band we obtain a binary fraction of 100-107 per cent. The two numbers should be the same and the discrepancy is likely due to small-number statistics. Finally, we note how the previously derived short-period PN binary fraction of 15-20 per cent is far larger than expected based on the main-sequence binary fraction and period distribution. As a byproduct of our analysis we present an accurately vetted compilation of observed main-sequence star magnitudes, colours and masses, which can serve as a reference for future studies. We also present synthetic colours of hot stars as a function of temperature (20-170 kK) and gravity (log g = 6-8) for Solar and PG1159 compositions.