NFL's Telvin Smith Gave Underage Girl $100 After Sex, Officials Say

7:06 AM PT -- Telvin Smith allegedly had two sexual encounters with an underage girl -- and officials say they have everything from DNA to audio recordings to prove his guilt.

In court documents, obtained by TMZ Sports ... officials spell out their case against 29-year-old Smith -- and it's pretty damning.

Officials say Smith met the girl at her work back in August 2019 ... and he pursued her hard, despite the girl's co-workers saying it was clear to the NFL star she was under 18.

Officials say Smith had sex with the girl just days later ... and offered her $200 for her troubles. The alleged victim told authorities she declined the cash at first, but says Smith ultimately "pressured her into taking a $100 bill."

The alleged victim told officials she later used that money to buy emergency contraception.

According to the court docs, the alleged victim says Smith told her "that if anyone raised concerns about them speaking, to say that Smith was serving as a 'mentor' to her."

The alleged victim claims there was a second incident in September 2019, when Smith picked her up in his Cadillac Escalade and they had sex inside the vehicle in a nearby parking lot.

The alleged victim told family members about the incidents -- and police were notified.

In November, investigators executed a search warrant and seized Smith's Cadillac Escalade to collect DNA evidence. Officials insinuate in court docs, DNA collected in the car matches the alleged victim.

There's more ... officials say the alleged victim also has voice memos from Smith in which he calls her "baby," requests that she spend the night, and makes kissing noises to her.

As we previously reported ... Smith was arrested Wednesday afternoon for unlawful sexual activity with certain minors. Records show he posted bond and was released late Wednesday night.

3:20 PM PT -- Telvin is smiling from ear to ear in his mug shot ... so it appears he's not too worried about the arrest.

3:16 PM PT -- Smith entered the Duval County Jail in cuffs ... wearing red and black basketball shorts and a black t-shirt -- new footage shows.

Smith reportedly turned himself in ... and, afterward, WJXT's Vic Micolucci reports law enforcement officials -- including SWAT and CSI technicians -- combed through his Florida home.