Being eternal means "not having precedence  of existence in the past".¹That is to say its inexistence is not supposed in the past and in the first place. Also eternity means a state of immortal existence  in future.². So when we talk about being eternal or immortal we should be mindful of the exact meaning of these terms. If you pay a little attention to the meaning of these terms you will come to know that Satan's ask for respite to be alive till the Day of Judgment does not imply being eternal and immortal.
 
Satan's Request for Respite
Satan is God's creature and each creature is under control of his creator. Therefore Satan cannot make himself involved in God's affairs volitionally. In fact, like other creatures whatever he has belongs to God. Upon getting expelled from divine kingdom due to his wrong deeds and arrogance, Satan asked God for more time to  mislead mankind from the right path because he was jealous of man. God also accepted his request to test man. The story of Satan's expulsion from the kingdom of God is mentioned in many verses of the Quran one of which will be mentioned so as to make clear the truth of the discussion.
These points have been referred to in verses 38 to 42 of Chapter al-Hijr. After Satan’s expulsion God says:"Curse upon you till the Day of Judgment!"³ Satan expresses some rude words to God which He does not accept absolutely and entirely.
 
1-Satan says: My Lord, grant me a reprieve (let be alive) until the Day they are resurrected.”Day of Judgment.⁴
2-Iblis has limited God’s servants to sincere and purified believers and has refuted his control over them and he considers the rest of human beings upon whom he has control  not to be God's servants. ⁵
3-The accursed Satan claimed independence in misleading mankind (as it is clear from surface meaning of the sentence" لاغوينهم"  (to mislead) that is used to signify conflict with God and revenge). God, the Sublime, rejected Satan's claim and called his assumption as invalid.⁶
God answers to the first request: God gives time to Satan but not as much as he asks and God says: "You are among those to whom we gave a certain amount of time" (but not until the Day of Judgment).⁷
Late Allamah Tabatabai  says: "This sentence is the God’sanswer to Satan in which a part of his request is rejected and another is accepted. The accepted part is related to keeping him alive and the rejected part is related to its duration till the Day of Judgment that is just for a fixed time not until the Day of Judgment. According to the form of both verses in question it seems too clear that "fixed day" is different from the "Day of Judgment".