Single-walled carbon nanotubes (SWCNTs) are powerful building blocks for near-infrared (NIR) fluorescent biosensors. They can be chemically tailored to detect specific biomolecules. Their performance in (bio)imaging applications depends on how fast analytes bind and unbind and change their fluorescence but measuring fast kinetics remains a challenge. Here, we demonstrate single-molecule detection by single SWCNT-based nanosensors and measure their rate constants. We use fluorescence microscopy (>900 nm) to image DNA functionalized (6,5)-SWCNTs, which detect the neurotransmitters dopamine and epinephrine as well as the interfering molecule ascorbic acid. By analyzing fluorescence traces of these sensors, we directly observe binding and unbinding events of individual analyte molecules. Hidden Markov modeling allows us to obtain dwell times and consequently rate constants (k_off). Debinding is best described by two exponential first order kinetics with k_(off,fast)=0.40 s^(-1)-0.71 s^(-1) and k_(off,slow)=0.01-0.10 s^(-1) depending on DNA sequence and analyte. Additionally, we find that sensitivity is correlated with sensor brightness and darker sensors have faster off-rates. We also show that differences in rate constants can be exploited for kinetically improved sensor selectivity (KISS) beyond the thermodynamic limit. Overall, we provide fundamental insights into the kinetics and mechanism of SWCNT-based sensors and propose a concept for kinetic selectivity.