No, we don’t. The classic example in the Bible of someone who became a Christian is the thief on the cross (Luke 23:40-43). There was no way he could be baptised, as he was being crucified at the time, and Jesus still says to him, “Today you will be with me in paradise.”. Baptism is just a symbol. Its a picture of what becoming a Christian means (washing/cleansing/fresh start/being buried and raised/etc) but it’s no more than a picture. In one way it’s a bit like having a badge saying that you are a doctor. You can be a doctor without having the badge, and you can have the badge without being a doctor. You can be a Christian without being baptised, and you can be baptised without being a Christian. Its just a sign.
Becoming a Christian isn’t about anything that we do. It’s about what Jesus has done for us. He died instead of us, he took the punishment that should have been ours so that we won’t have to be punished. He offers to deal with all the wrong that we have done. All we have to do is (to use a bible phrase) repent and believe. Believe just means trust or rely on what Jesus has done to rescue us - seeing that it’s true and saying thank you. Repent means turn around - the reason God is angry with us is because we’ve gone our own way instead of his. Becoming a Christian means recognising that. And recognising that Jesus has dealt with it. It means turning from going our own way, and starting to live with God as God. Baptism doesn’t have any part in this. All baptism is, is a physical picture of part of what is going on, pretty much equivalent to the verbal picture that I’ve tried to paint just now. Baptism doesn’t have anything to do with becoming a Christian.
Having said that, the expectation in the Bible is that Christians will get baptised. I guess it’s a bit like the questions, Do you have to read the Bible to be a Christian? Do you have to go to church to be a Christian? No. But if someone is a Christian then it would be a bit strange if they didn’t do those things. Its perhaps a little bit like saying, “Do you have to take your clothes off to have a shower?” Well, no. But it would be a bit odd if you didn’t. It’s the same with baptism - it would be odd for a Christian not to get baptised. But they would still be a Christian.