After signing a professional contract with this club, he was loaned to other Erovnuli Liga teams for three seasons before his transfer to Valencia.

[7] On 7 June 2021, Mamardashvili joined La Liga club Valencia on a one-year loan deal, with a buyout clause, and was initially assigned to the reserves in Tercera División RFEF.

[8][9] He made the pre-season with the main squad, impressing manager José Bordalás[10] and earning himself the starting spot in the league opener against Getafe on 13 August, which ended 1–0 for Los Che.

Overall, during his third season at the club Mamardashvili continued to achieve impressive results, namely, improving his clean sheet record (13), saving the most penalties (3) along with Unai Simon and preventing the most goals in La Liga.

On 27 August 2024, Valencia and Premier League club Liverpool confirmed that Mamardashvili would be joining the latter at the start of the 2025–26 season.

[24] For his performance in the return leg (2–0) in September, Mamardashvili was named in Team of the Week after round 5,[25][26] and whoscored.com included him in monthly and seasonal Best XI of League C.[27] In the summer of 2023, Mamardashvili took part in three games in UEFA European Under-21 Championship for Georgia U21s, who had a highly successful group stage campaign and reached quarter-finals for the first time in their history.

[28] On 26 March 2024, Georgia defeated Greece 4–2 in a penalty shoot-out in the play-off final to qualify for UEFA Euro 2024, the nation's first ever major international tournament.

[31] Analytical sites such as Sofascore, Marca and 90min.com named him in the Team of match day 2, with the former awarding him the second highest aggregate rating point after Spanish midfielder Fabián Ruiz.

[32][33][34] In the team's third Group F match, Mamardashvili kept a clean sheet against Portugal, as Georgia won 2–0 to qualify for the knockout stage.