Not exactly. A gift is just something given to someone without payment.
You doing what you signed up for in the New Covenant (yes there is a literal covenant and there are conditions) is not doing anything beyond the call of duty and reasonable service. You work, God reaps the glory, and you’re entitled to no pay, but gifts and rewards are many. They often referred to it as being a “slave” of righteousness.
You seek God’s glory, serve Him, and there are rewards and gifts.
It being a gift, it isn’t something you’re entitled to. Make more sense that way?
1st, If you work for it, it isn't a gift.
Rom 4:
2 For if Abraham were justified by works, he hath whereof to glory; but not before God.
3 For what saith the scripture? Abraham believed God, and it was counted unto him for righteousness.
4 Now to him that worketh is the reward not reckoned of grace, but of debt.
5 But to him that worketh not, but believeth on him that justifieth the ungodly, his faith is counted for righteousness.
(MY NOTE: Justification/Righteousness/Salvation/Eternal Life are gifts given via Faith, Not works)
2ndly, No place have I claimed or posted, that a believer shouldn't serve the Lord or that salvation is an entitlement. Post claims its a free gift.
Holy Spirit - A gift given thru faith: Acts 10:44-45, 11:17
Eternal Life - A gift given thru faith: Rom 6:23, Jn 3:36. 5:24, 6:47
Righteousness - A gift given thru faith: Rom 3:24 4:3 & 5, 5:17-18
Grace - A gift given thru faith: Rom 3:24, 5:15, Eph 2:8-9
Justification - A gift given thru faith: Rom 3:28, 4:5, 5:1
After the gifts of grace/righteousness/justification/salvation/eternal life are given by the Lord. All solely based on Faith placed in/on the merits/work of His sin atoning death & resurrection. All/any work done serving the Lord will receive a payment/reward (1 Cor 3:8-15). Even a drink of water given in His name (Matt 10:42) will receive a reward.