Bangladesh found themselves in trouble at 49/4 but Liton and Rahim's unbeaten 204-run partnership resurrected the hosts and they finished the first day on top after a bleak morning session.
Bangladesh have been in great T20I form lately, first winning 2-1 away from home against Zimbabwe and then routing Australia 4-1 at home.
Rahim, who plays for Abahani Limited in the Dhaka Premier League T20 tournament, sustained the fracture during his team's match against Gazi Group Cricketers on Monday.
Rahim, a former skipper, asked spinner Mehidy Hasan Miraz to obstruct Sri Lanka No. 3 batsman Pathum Nissanka during running between wickets.
Mushfiqur Rahim was in cracking form for the hosts, slamming a 127-ball 125 to get his team to 246 on a tricky wicket after Bangladesh won the toss and elected to bat.
Sri Lanka were all out for 224 in reply to Bangladesh's 257/6.
Following his on-field behaviour, Mushfiqur was also fined 25 per cent of the match fee.
The incident happened in the 17th over of the second innings when the opposition, Fortune Barishal, needed 45 runs off 19 deliveries.
Bangladesh star wicketkeeper-batsman Mushfiqur Rahim was involved in an ugly incident during a recent Bangabandhu T20 Cup encounter between Beximco Dhaka and Fortune Barishal.
The incident took place in the 17th over of the second innings when the Barishal side needed 45 runs off 19 balls.
The Bangladesh Cricket Board (BCB) has prepared four venues for the first phase of practice session by following all the necessary safety protocols, according to a BCB statement.
The BCB Chief Executive said that the board will follow the ICC guidelines regarding the resumption of team training.
Rahim had earlier placed his name for the IPL 2020 auction and was even party of the final 338-man list. He had kept his base price as INR 75 lakhs, but went unsold.
Former Pakistan all-rounder Shahid Afridi bought Mushfiqur Rahim's bat for USD 20,000, which was put up for auction to raise funds for coronavirus relief.
Mushfiqur Rahim has decided to auction his bat, with which he scored a double hundred against Sri Lanka in 2013.
Bangladesh Cricket Board president Nazmul Hassan said that wicketkeeper and star batsman Mushfiqur Rahim should go to Pakistan for the one-off ODI and second Test in Karachi in April
Mushfiqur Rahim scored a double century in Bangladesh's innings and 106-run win over Zimbabwe in the one-off Test.
The right-arm off-break bowler was well supported by Taijul Islam who returned figures of 4/78 as the visitors were all out for 189 in 57.3 overs in their second innings on Day 4 of the Test match.
Rahim registered his third Test double century on Monday and now has scored 4,413 runs in 70 Tests, which include seven hundreds and 21 half-centuries.
With his double century, Rahim became the highest run-scorer for Bangladesh in Test cricket with 4,412 runs, seven more than Tamim Iqbal's tally of 4,405 runs.
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