K-ion batteries attract extensive attention and research efforts because of the high energy density, low cost, and high abundance of K. Although they are considered suitable alternatives to Li-ion batteries, the absence of high-performance electrode materials is a major obstacle to implementation. On the basis of density functional theory, we systematically study the feasibility of a recently synthesized C6BN monolayer as anode material for K-ion batteries. The specific capacity is calculated to be 553 mAh/g (K2C6BN), i.e., about twice that of graphite. The C6BN monolayer is characterized by high strength (in-plane stiffness of 309 N/m), excellent flexibility (bending strength of 1.30 eV), low output voltage (average open circuit voltage of 0.16 V), and excellent rate performance (diffusion barrier of 0.09 eV). We also propose two new C6BN monolayers. One has a slightly higher total energy (0.10 eV) than the synthesized C6BN monolayer, exhibiting enhanced electronic properties and affinity to K. The other is even energetically favorable due to B-N bonding. All three C6BN monolayers show good dynamical, thermal, and mechanical stabilities. We demonstrate excellent cyclability and improved conductivity by K adsorption, suggesting great potential in flexible energy-storage devices.