The idea of turning a drill press into a small lathe is not new. The setups that you may find in the internet include
The work of the instructables contributor Tool Using Animal (https://www.instructables.com/id/Drill-Press-Lathe/)
A few youtube videos.
A commercial product under the name of Vertilathe
These setups are similar in principle. Specifically the commercial product gave me the idea to go ahead and make a simple setup out of wood as a first try, more than 5 years ago. It turned out that it works well and I did not have to modify it for the simple projects I am doing. This is what I am presenting here. It consists of a unique block that contains the two basic components of the lathe, the tool rest and the live centre.The tool rest
This is a vertical block of wood 12 cm x 5cm x 1.5cm firmly screwed and glued on the basis. It serves to slide the tool up and down along the edge of the working piece. Furthermore it protects your hand from getting to close to the turning piece. The length depends upon the available space you have in the drill press. In retrospect I should have made this a few centimeters longer.The live centre
This allows the working piece to rotate around a stable axis. It consists of a short axis with a conical shape usually mounted on a ball-bearing. I made mine in the simplest possible way and it proved enough. It is made out of a screw free to rotate in its hole, supported by double nuts and washers.