If only I had a proverbial penny for every time I have heard “I can’t play without music” in my time playing. Like many things in life memorising music is just a practised skill and pretty much everyone CAN do it. Those that claim they can’t almost certainly have never attempted to learn how and for some reason it is one of those things that scare many players.
OK so how can you learn, baby steps is the easiest answer. Find a tune you are reasonably proficient playing when reading. Play only the first two bars of the piece whilst reading the music. If you played it correctly then turn the music away and try again without the music. If you played incorrectly then repeat the process. Once the first two bars have been mastered without music repeat the process with bars three & four. Once this is mastered bolt the first two bars to the second two. Voila you can play four bars without music. Yes I know its obvious but as a student have you thought about doing this before. Over the next days continue the process by adding the following 2 bars until you get to the end of the end of the song, have patience practise in small chunks it will come!
So why bother to learn this skill I hear you ask. Well if you are not watching the written music you can concentrate on other things like playing the right notes, such as putting feeling into the music you are playing and even down to the simple pleasure of enjoying the music emanating from your instrument. Enjoy!