Does a Christian need to continue repenting?

In one sense, your statement is correct. The Bible teaches that a person need not ask for forgiveness of sins more than once to be saved. Once we have appealed to God for His mercy, having trusted in His Son to save us from the penalty of sin, we are saved eternally and will never bear the punishment for our sins. As Paul summarizes

Rom. 10:9: "that if you confess with your mouth Jesus as Lord, and believe in your heart that God raised Him from the dead, you will be saved..."

Nothing more is required.

On the other hand, a Christian is also called by Scripture to maintain a repentant heart by routinely acknowledging sin and seeking God's grace and mercy in the face of our daily stumbles into sin. This type of repentance is not done for the purpose of obtaining or maintain our salvation, yet it is required nonetheless.

In other words, there are two different types of repentance in the Bible, and VBVMI uses a simple convention to distinguish between them. We could say the Bible teaches Repentance (capital "R") and repentance (lowercase "r"). Repentance (capital "R") refers to the repentance that accompanies our salvation moment. This is a turning away from dead works and our previous life lived apart from God and an acknowledgment that we are sinners and in need of God's mercy and forgiveness. Simply put, this is the repentance unto salvation. 

This type of repentance is a unique, one-time event prompted by the Holy Spirit in the heart of an unbeliever, which leads to saving faith in the Gospel, as Paul states:

For the sorrow that is according to the will of God produces a repentance without regret, leading to salvation, but the sorrow of the world produces death. - 2Cor 7:10

The second kind of repentance (lowercase "r") is the conviction believers experience daily when made aware of their sinful behaviors. Becoming a Christian removes the penalty of sin, but it doesn't (immediately) remove the reality of sin in our earthly life. So, we receive this secondary form of repentance by the work of the Holy Spirit to encourage our sanctification. As the Holy Spirit convicts us of our sin, He prompts our repentance which leads us toward a more Christ-like life.

This second kind of repentance isn't an appeal for salvation, since we already possess salvation by faith. Rather, it is an appeal for God to remove any earthly consequences of our sin and to encourage us into a better walk of obedience. We are appealing for relief from the discipline of the Lord much like a son appeals to his human father for mercy to avoid punishment. And John says the Lord will forgive us in this way if we repent:

1John 1:9  If we confess our sins, He is faithful and righteous to forgive us our sins and to cleanse us from all unrighteousness. 

There are a couple of good examples (among many) in Scripture to illustrate this principle of repentance for the believer. First, consider Jesus' instructions to the church of Ephesus in Revelation:

Rev. 2:5 ‘Therefore remember from where you have fallen, and  repent and  do the  deeds you did at first; or else I am coming to you and will remove your  lampstand out of its place — unless you repent.

Jesus told the church in Ephesus that they must repent of their poor witness (i.e., leaving their first love), or else He will end the church's very existence. Here we see a connection between a believers' sin and God visiting the consequences of their sin upon them. If the church didn't repent of their sin, Jesus would visit the consequences of their sin upon the church.

Likewise, if believers live in unrepentant sin, they will see the consequences of their sin visited upon them as discipline from the Lord. As the writer to Hebrews 12:7-11 says plainly:

It is for discipline that you endure; God deals with you as with sons; for what son is there whom his father does not discipline?
But if you are without discipline, of which all have become partakers, then you are illegitimate children and not sons.
Furthermore, we had earthly fathers to discipline us, and we respected them; shall we not much rather be subject to the Father of spirits, and live?
For they disciplined us for a short time as seemed best to them, but He disciplines us for our good, so that we may share His holiness.
All discipline for the moment seems not to be joyful, but sorrowful; yet to those who have been trained by it, afterwards it yields the peaceful fruit of righteousness.

A second example is found in the life of David in 2Sam 12:13-16:

Then David said to Nathan, “ I have sinned against the LORD.” And Nathan said to David, “The LORD also has taken away your sin; you shall not die.
“However, because by this deed you have given occasion to the enemies of the LORD to blaspheme, the child also that is born to you shall surely die.”
So Nathan went to his house.  Then the LORD struck the child that Uriah’s widow bore to David, so that he was very sick.
David therefore inquired of God for the child; and David fasted and went and lay all night on the ground.

In 2Sam 12, David is praying and fasting for the Lord to spare his son, who is near death. David knows that his son's condition was brought upon him by the Lord as punishment against David for his unrepentant sin with Bathsheba. David prays for the Lord to show mercy instead. The Scripture says plainly that David's sin has already been taken away by his faith, so salvation was not in question. Nonetheless, David feels the need to pray for forgiveness, because he wishes to save his son.

The Lord brought discipline against David because of his sin, and even knowing he had been forgiven, David still prayed for mercy for his son's sake. Obviously, the Lord rejected David's plea and took the son anyway, but this is an excellent example of why a believer should pray in repentance for forgiveness even though our salvation is secure.

Clearly, the Lord expects believers to pray for forgiveness even after we are saved, but we do it for a different reason than the one we had before faith. Before faith, we prayed with a repentant heart longing for forgiveness leading to eternal life. Now having been saved, we still pray for forgiveness from the consequences of our sin.

Scripture quotations taken from the (NASB®) New American Standard Bible®, Copyright © 1995, 2020 by The Lockman Foundation. Used by permission. All rights reserved. www.lockman.org