I search for songs, that have a note, that stays through the whole song?
My plan is to put together a playlist with popular songs with at least one song for each note to train your perfect pitch.
In Western music there are many examples with a note that is kept for a long time but usually a change happens sooner or later. On a piano or giutar it is a repeated note since you can not sustain a note on those instruments. It is usually referred to as a pedal ponit.
You want popular songs, so I am not sure what you think of the following pieces. At least for some people they are popular.
Chopin's Rain Drop Prelude for piano. It has a constant repeated A♭, sometimes notated as G♯. There are a few bars where the repeated note changes to another one, but only for a short while, after which the music goes back to the repeated A♭/G♯. Here is an image from the piece:
Stravinsky's Augurs of Spring (from the The Rite of Spring) has one chord that is repeated during the entire movement. Here it is on YouTube: Augurs of Spring