No, they do not. Neutrinos are weird, but they aren’t that weird.
The origin of this misconception comes from a 2011 result. The OPERA experiment data showed neutrinos arriving at the detector surprisingly quickly, supposedly traveling faster than the speed of light. Other experiments in the same neutrino beam (and elsewhere around the world) were unable to replicate the anomaly. The OPERA team later discovered a faulty piece of equipment (a cable) was responsible for the timing mismatch. Once repaired, OPERA also clocked neutrinos as very close to the speed of light, but not surpassing it.