During the study of soot formation in laminar jet flames of methane at elevated pressures in a high-pressure combustion chamber, we have observed an anomalous occurrence at the chamber pressures of 6 MPa and higher. At pressures between 6 and 9 MPa, after the laminar methane jet flame had been stabilized on a co-flow circular nozzle-type burner, one of the following occurrences was observed: (a) the jet flame is completely extinguished without any external interference, and immediately after, a liquid material started to flow out of the fuel nozzle, and (b) the jet flame started flickering or changing its shape, and at the same time, a liquid stream started trickling out of the nozzle. This outflowing liquid influenced the jet flame by widening the flame near its base and decreasing the visible flame height. In consideration of the temperatures and the pressures involved, the liquid is believed to be a mixture of methane and water formed by the water vapor in the combustion products of the flame mixing with supercritical methane inside the nozzle.