The Bible describes Christians being called away when Jesus returns. Two particularly favorite verses are in Matt 24:40-41 and 1Thess 4:16-17.
Then two men will be in the field; one will be taken and one left. Two women will be grinding at the mill; one will be taken and one left. (Matt 24:40-41)
For the Lord himself will descend from heaven with a cry of command, with the voice of an archangel, and with the sound of the trumpet of God. And the dead in Christ will rise first. Then we who are alive, who are left, will be caught up together with them in the clouds to meet the Lord in the air, and so we will always be with the Lord. 1Th 4:16-17
So there will be a “rapture” in the sense that people will be called away when Jesus returns — something to really look forward to. But there’s a lot that isn’t clear, from the fun questions like, “what if you’re driving when it happens?” through to the very much more serious questions like “what is the sequence of events at that time”?
The Bible describes repeated cycles of gathering, judgment, redemption, subjugation of the world to God’s will. There’s a number of tantalizing verses that hint at what might happen, such as in Revelation 20:
Then I saw thrones, and seated on them were those to whom the authority to judge was committed. Also I saw the souls of those who had been beheaded for the testimony of Jesus and for the word of God, and those who had not worshiped the beast or its image and had not received its mark on their foreheads or their hands. They came to life and reigned with Christ for a thousand years. The rest of the dead did not come to life until the thousand years were ended. This is the first resurrection. Blessed and holy is the one who shares in the first resurrection! Over such the second death has no power, but they will be priests of God and of Christ, and they will reign with him for a thousand years. Rev 20:4-6
There are a lot of different ways to understand the many verses about events around Jesus’ return, and many different ways to tell the story. So any answer to the question has to be tentative. But almost all sequences of events I’ve ever read have the first resurrection here after the “rapture” (usually immediately after). And it’s clear that there is a second resurrection:
And I saw the dead, great and small, standing before the throne, and books were opened. Then another book was opened, which is the book of life. And the dead were judged by what was written in the books, according to what they had done. And the sea gave up the dead who were in it, Death and Hades gave up the dead who were in them, and they were judged, each one of them, according to what they had done. Then Death and Hades were thrown into the lake of fire. This is the second death, the lake of fire. And if anyone’s name was not found written in the book of life, he was thrown into the lake of fire. (Rev 20:12-15)
So the answer is that, probably, yes, people will be saved after the rapture of the church, if the rapture of the church is that described above. But much of what is commonly written about the rapture of the church is not based on scripture — be alert: check what they say against the Bible; much of it is just wishful thinking.