The Griffin powered Mk XIX was the fastest of the Spitfires, this was used to dog fight the Lightning in 1963 as mentioned above, the potential ( or actual) enemy operated P51Ds.
The first US pilot to shoot down a jet was Chuck Yeager flying a P51D, he shot down a ME262.
The RAF did shoot down a few ME262s operating Spitfires, 14/2/1945 Flight Lieutenant FAO Gaze of RAF 610 Sqn shot down a ME262, and in March 1945 Flying Officer Nicholson also shot one down over Germany.
Although the ME262 was faster than the Spitfire, tests demonstated that the ME262 would lose control at Mach 0.86 and the airframe would break up shortly afterwards, the Spitfire could better this in combat.
Source is " Spitfire" by Jonathan Glancey, a very interesting read on the subject.
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om15