Are there bridge rectifier chips that work at 40 kHz?
I would just build your bridge rectifier out of discrete diodes; that gives you many more options:
Signal diodes tend to have better recovery characteristics than power diodes, in general. It's possible something as simple as 1N4148s will server your purposes.
Schottky diodes have no recovery time at all, due to being unipolar devices, but have significant reverse leakage and low blocking voltages. They also have a lower forward voltage than silicon pn diodes.
FREDs (fast recovery epitaxial diodes) have all the advantages of standard pn diodes with low reverse leakage and high blocking voltage, but they have recovery times in the tens of nanoseconds due to a light doping of either platinum or gold, which both act to shorten carrier lifetime.
Silicon carbide schottky diodes still have the zero recovery time that schottky diodes are known for, but due to the higher bandgap of SiC they can have extremely high blocking voltages and low reverse leakage--superior even to silicon pn diodes. They're pricier than silicon diodes, but still affordable until you start needing truly absurd specs. Note that the forward voltage of a SiC schottky is typically higher than a silicon pn diode, so you do lose out a bit in that regard. You can mitigate this somewhat by using diodes rated for much higher current than what you intend to use them at.
PIN diodes are also used for RF purposes, but I'm not as familiar with those as the other types on this list. They do have lower carrier lifetimes, so they may meet your requirements.
I have no idea about the recovery characteristics of vacuum tube diodes, but I would think they're probably pretty good, since they're also unipolar devices... but I strongly doubt their advantages would even come close to outweighing the downsides (need to power the heaters, which wastes energy; need to replace them periodically; expensive; hard to get....).