A common allegation of Catholic and Orthodox apologists is that their church is the original church. It goes straight back to Christ, whereas Protestant churches are upstarts. These didn't pop into existence until the 16C.
One problem with that allegation is that it's only as good as your paradigm of the church. Put it this way, do you define the church by the vehicle or the passengers?
How does the NT describe the church? As the community of faith. A fellowship of believers, and families of believers. They are united by the grace they share and their common faith in the message of Scripture. In addition, there's a minimal polity (elders, deacons) and at least two sacraments (baptism, communion).
Basically, the NT defines the church by the passengers, not the vehicle. If the identity of the church is centered on the passengers rather than the vehicle, then the church can exist continuously even if the vehicle changes, just as passengers can change vehicles, but remain the same passengers.
The church as passengers goes back to NT times. Indeed, God has always had a community of faith. Protestants can trace the church back as far as you please. In the Reformation, they changed cars.