The origin of the Li-rich K giants is still highly debated. Here, we investigate the incidence of binarity among this family from a nine-year radial-velocity monitoring of a sample of 11 Li-rich K giants using the HERMES spectrograph attached to the 1.2m Mercator Telescope. A sample of 13 non-Li-rich giants (8 of them being surrounded by dust according to IRAS, WISE, and ISO data) was monitored alongside. When compared to the binary frequency in a reference sample of 190K giants (containing 17.4% of definite spectroscopic binaries - SB - and 6.3% of possible spectroscopic binaries - SB?), the binary frequency appears normal among the Li-rich giants (2/11 definite binaries plus 2 possible binaries, or 18.2% SB + 18.2% SB?), after taking account of the small sample size through the hypergeometric probability distribution. Therefore, there appears to be no causal relationship between Li enrichment and binarity. Moreover, there is no correlation between Li enrichment and the presence of circumstellar dust, and the only correlation that could be found between Li enrichment and rapid rotation is that the most Li-enriched K giants appear to be fast-rotating stars. However, among the dusty K giants, the binary frequency is much higher (4/8 definite binaries plus 1 possible binary). The remaining 3 dusty K giants suffer from a radial-velocity jitter, as is expected for the most luminous K giants, which these are.