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For reference, the rule on Critical Hits says (PHB, p. 196):

When you score a critical hit, you get to roll extra dice for the attack’s damage against the target. Roll all of the attack’s damage dice twice and add them together. Then add any relevant modifiers as normal. [...] If the attack involves other damage dice, such as from the rogue’s Sneak Attack feature, you roll those dice twice as well.

There are a number of creatures in the game which do a fixed amount of damage on a hit (rather than rolling a damage die). Do these attacks do extra damage on a critical hit?

One example is the Sprite:

Longsword. Melee Weapon Attack: +2 to hit, reach 5 ft., one target. Hit: 1 slashing damage.

Since no die is rolled, does the sprite deal 1 or 2 points of damage on a critical hit? Rules as Written based answers are preferred (and seem to support not doubling), but a developer ruling or statement would be much appreciated.

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Only Damage Dice Are Rolled Twice.

You quoted the relevant section:

Roll all of the attack’s damage dice twice and add them together.

That is the rule.

In order to add anything else to the damage of a critical hit, you would need to find an exception to it - such as the one found in the description of the legendary magic item Wave (DMG, p. 218):

If you score a critical hit with it, the target takes extra necrotic damage equal to half its hit point maximum.

As it stands, you are rolling no damage dice to deal the 1 damage, so "rolling all of the attack's damage dice twice" causes no change in the damage dealt.

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