We know that drivers give maximum performance during qualifying race without worrying about overheat engine and have lighter fuel. Also, based on 2016 regulation, drivers can open the DRS wing anytime they want during qualifying. So, based on this factor, I can sure say that qualifying time will be much faster than fastest lap time. But can I know is there a F1 race where the fastest lap is quicker than pole position?
Races where qualifying is held in wet conditions and the race is dry are the obvious answer, due to the significant time differential between wet and dry conditions. Most recently the 2022 British Grand Prix had this occur - Pole was 1:40.983 for Carlos Sainz in the wet and the fastest lap in the race was a 1:30.510 for Lewis Hamilton in the dry.
Similarly the 2021 Russian Grand Prix saw a Pole of 1:41.993 for Lando Norris who then set a fastest lap of 1:37.423 in the race. His pole lap was set on slick (dry weather) tyres but the track had been wet earlier in the session and was drying rather than fully dry.
I could go on, but pretty much any time qualifying is wet and the race is dry you'll see that happen.
But has it ever happened with a dry qualifying? I had a look through the last 2.5 seasons as I wondered whether the relatively recent change to awarding a championship point for fastest lap (and the resultant occurrences where people have pitted very late in the race to get it) might have resulted in this happening but it seems that although races where this occurs results in a fastest lap much closer to the pole time there's still a second or two in it. It has happened however - at the 1957 German Grand Prix Juan Manuel Fangio set pole position with 9:25.6 and went on to set a fastest lap of 9:17.4 in the race.
Motosubatsus Answer has some interesting races that had weather changes between the qualifying and the race but there were races in which the fastest on track lap was set in the same weather as the pole positon:
Austria 2003 for example tho it was noteworthy for uncertain weather conditions so they did use different tyres meaning pole wasn't the fastest qualifying time.
Similiarly the 2005 Turkish Grand Prix had excellent weather on both race day as well as Qualifying
Lastly the 2005 Chinese Grand Prix had excellent weather on both race day as well as Qualifying German source for the qualifying weather with both the pole and the fastest qualifying time being slower than the fastest lap time.
Apparently Australia 2004 was the same, tho I could not verify the weather.
China in 2005 was the only one of these which I really consider as having the exact same conditions between race and qualifying tho.