Is it permissible for an employee to claim that their manager is absent or busy at the manager's request to avoid unwanted visitors, and what is the Islamic ruling on this, considering the risk of losing one's job if one refuses?

Lying is not permissible and is forbidden by definitive proofs, such as the Almighty's saying: "And we make the curse of Allah upon the liars," and the Prophet's ﷺ saying: "Can a believer be a liar? He said: No." Informing someone that the manager is not present when he is, is plain lying. However, informing them that he is busy is an easy matter, as it applies to any state of being busy. If the manager insists on being told that he is not present, one can imply by saying: "He is not here," meaning he is not in front of the speaker or in his room. However, this should only be resorted to when there is a fear of injustice, such as being fired from work.