No, a person cannot be saved without the Holy Spirit. You cannot grow as a Christian, and enter into a close personal relationship with Jesus, without the Holy Spirit. The Holy Spirit convicts us of sin, He leads us into all truth, He lives in us to lead us to Jesus, He seals us for salvation. The problem is that there are many Christians who live their whole lives in the church, think that they are saved, but are not. This is because they have not received the Holy Spirit and therefore do not enter into a relationship with Jesus, even though they were baptized, and they are ultimately lost.

Look at what happened with the apostle Paul and some Christian people he met.

Acts 19     And it happened, while Apollos was at Corinth, that Paul, having passed through the upper regions, came to Ephesus. And finding some disciples ² he said to them, “Did you receive the Holy Spirit when you believed?”

So they said to him, “We have not so much as heard whether there is a Holy Spirit.”

³ And he said to them, “Into what then were you baptized?”

So they said, “Into John’s baptism.”

⁴ Then Paul said, “John indeed baptized with a baptism of repentance, saying to the people that they should believe on Him who would come after him, that is, on Christ Jesus.”

⁵ When they heard this, they were baptized in the name of the Lord Jesus. ⁶ And when Paul had laid hands on them, the Holy Spirit came upon them, and they spoke with tongues and prophesied.

Here were people who were Christians, but did not know of the Holy Spirit, and had never received Him. If anyone is unsure if the Holy Spirit lives in them, and leads them, Jesus said that they should ask the Father to give them the Spirit.

Luke 11:13   “If you then, being evil, know how to give good gifts to your children, how much more will your heavenly Father give the Holy Spirit to those who ask Him!”

Here are some passages that show us what the Holy Spirit does for us, to bring us to salvation and eternal life.

Mark 13:11   But when they arrest you and deliver you up, do not worry beforehand, or premeditate what you will speak. But whatever is given you in that hour, speak that; for it is not you who speak, but the Holy Spirit.

John 14     ¹⁶ And I will pray the Father, and He will give you another Helper, that He may abide with you forever— ¹⁷ the Spirit of truth, whom the world cannot receive, because it neither sees Him nor knows Him; but you know Him, for He dwells with you and will be in you.

John 16     ⁷ “Nevertheless I tell you the truth. It is to your advantage that I go away; for if I do not go away, the Helper will not come to you; but if I depart, I will send Him to you. ⁸ And when He has come, He will convict the world of sin, and of righteousness, and of judgment: ⁹ of sin, because they do not believe in Me; ¹⁰ of righteousness, because I go to My Father and you see Me no more; ¹¹ of judgment, because the ruler of this world is judged. ¹² I still have many things to say to you, but you cannot bear them now. ¹³ However, when He, the Spirit of truth, has come, He will guide you into all truth; for He will not speak on His own authority, but whatever He hears He will speak; and He will tell you things to come.

¹⁴ He will glorify Me, for He will take of what is Mine and declare it to you. ¹⁵ All things that the Father has are Mine. Therefore I said that He will take of Mine and declare it to you.”

Romans 8: 26     Likewise the Spirit also helps in our weaknesses. For we do not know what we should pray for as we ought, but the Spirit Himself makes intercession for us with groanings which cannot be uttered.

2 Peter 1:21    for prophecy never came by the will of man, but holy men of God spoke as they were moved by the Holy Spirit.

Ephesians 1     ¹³ In Him you also trusted, after you heard the word of truth, the gospel of your salvation; in whom also, having believed, you were sealed with the Holy Spirit of promise, ¹⁴ who is the guarantee of our inheritance until the redemption of the purchased possession, to the praise of His glory.

Galatians 5     ²² But the fruit of the Spirit is love, joy, peace, longsuffering, kindness, goodness, faithfulness, ²³ gentleness, self-control. Against such there is no law.

So, we see that the Holy Spirit is totally involved in our lives and ultimate salvation and eternal life, and we understand why Jesus sent the Holy Spirit to earth, after He was resurrected and went back to heaven.

I trust that this answers your question. God bless!