We demonstrate a cryogenic buffer gas-cooled molecular beam source capable of producing bright, continuous beams of cold and slow free radicals via laser ablation over durations of up to 60~seconds. The source design uses a closed liquid helium reservoir as a large thermal mass to minimize heating and ensure reproducible beam properties during operation. Under typical conditions, the source produces beams of our test species SrF, containing $5times10^{12}~$molecules per steradian per second in the $X^{2}Sigma(v=0, N=1)$ state with a rotational temperature of $1.0(2)~$K and a forward velocity of $140~$m/s. The beam properties are robust and unchanged for multiple cell geometries but depend critically on the helium buffer gas flow rate, which must be $geq10~$ standard cubic centimeters per minute to produce bright, continuous beams of molecules for an ablation repetition rate of $55~$Hz.
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