The left-handed opener sustained the injury from only the third ball he faced when a Suranga Lakmal delivery struck him.
Tamim briefly received treatment from the team physio but was forced to retire hurt on two not out at the Dubai International Stadium and is now expected to be out of action for six weeks.
The 29-year-old is Bangladesh’s all-time leading run-scorer and centurion in ODI cricket and his absence will come as a significant blow to the team’s hopes in the tournament.
The batsman returned to the crease having been struck on the hand and helped to add 32 for the final wicket in Bangladesh’s win over Sri Lanka.
The victory leaves Bangladesh well-placed with Sri Lanka and Afghanistan facing off in the second group B clash on Monday.
Two teams progress from the group stages to a stage called the Super Four. The other group is made up of Asian giants India and Pakistan and minnows Hong Kong who came through a qualifying tournament in Malaysia to earn their place.
Bangladesh were yet to name a replacement at the time of writing.