Not really
While there are some awesome .22 caliber rifles out there, there are none that have been adopted by anyone as a "sniper rifle". Most military sniper rifles out there shoot standard military calibers, and differ only in that they have ammunition which is more suited to thier task.
However, there are a number of varmint rifles out there in .223, .22-250, .220-swift, .22 hornet which are fine and accurate rifles, they don't meet the 'tacti-cool' mission of a 'sniper rifle'.
Accuracy for these calibers is excellent. I have fired quite a bit of match grade .223 in a Howa rifle I bought recently for proofing .223 loads. I have fired a number of savage 110 bolt guns in .22-250 that had astonishing accuracy. We were shooting at playing cards at 100 yards, and you could shoot the dagger out of the hand of the suicide king repeatedly.
For the most part however, these higher powered .22 cartridges are not really ideal for the role a sniper plays. Sniping is done at long range against human targets. .22-250 and .220-swift will pop a prarie dog, but the same bullets do not yield the kind of penetration required to take out a two-legged target at long range.
The .223 most closely fits this, being a military caliber, and having bullets that are designed to fulfill the sniper's role. However, most militaries classify .22 caliber rounds as being "too light" for the sniper role.
Until recently there was nothing truely approximating a sniper rifle chambered for the .223(5.56mm) cartridge. However, in the more recent war-on-terror the Seals, Army special forces, and others have now been issuing the SPR (Mk 12) to it's forces. This is essentially an AR-15 platform which has been optimized and accurized for the role of the designated marksman/sniper.
Other than this, there isn't much in the arena which fits the normal bolt gun (ala M40) approach to the sniper rifle.