## RAW *any* damage is a roll

As you have noted, a critical permits an extra roll of damage dice, but does not change any modifications to the roll.  That might make it seem like 'flat damage' is not augmented by a critical, but note that all damage is *defined as a die or dice*.  From the PHB on [Damage Rolls][1] (emphasis mine):
>Each weapon, spell, and harmful monster ability specifies the damage it deals. **You roll the damage die or dice**, add any modifiers, and apply the damage to your target. Magic weapons, special abilities, and other factors can grant a bonus to damage.

Damage *requires* a roll of a damage die, or dice, by definition. 
 Modifiers are qualified as 'any', indicating that they may or may not be present, but no such qualification is made for damage dice.  Damage is always a roll; we are **not** told 'You take any flat damage value, roll and add any damage die or dice, add any modifiers, and apply the damage to your target'.  For a hit with a sprite-wielded longsword that does a base of 1 damage, there is no RAW way to apply this amount as damage unless the value of 1 represents a roll.

For damage that is specified as a flat amount x, the damage die must be of the form xd1.  A sprite longsword (or blowgun needle, or [unarmed strike][2]) which does 1 point before the Strength modifier is thus a 1d1, whose only outcome is 1 damage.  On a critical the damage dice would be doubled to 2d1, or 2. 

At least, this is what we can conclude from the language in the PHB.  Is this what is intended?  Most likely not, and see below.  But it *is* what is written.

## RAI flat values are not rolls

As Jeremy Crawford [tweeted][3],
>An unarmed strike deals 1 + Str. modifier damage, even on a critical hit. 

The intention is that an unrolled damage amount is not increased on a critical.

## Roll20 treats flat damage values as rolls
If you use roll20, you may notice that a flat value is treated as a damage roll.  For example, if your character is equipped with a blowgun, the needle will do 1 point of damage (plus your Dex mod if you are proficient).  But on a critical hit, it will display the damage as 1 + 1. 

That the calculation of damage for a critical hit is applied in this manner in roll20 says nothing about RAW or RAI for the rule, but it is worth noting in case this application differs from the ruling at your table.


  [1]: https://www.dndbeyond.com/sources/basic-rules/combat#DamageRolls
  [2]: https://rpg.stackexchange.com/q/57051/23547
  [3]: https://twitter.com/JeremyECrawford/status/736085559597096960