I have been into many situations while converting designs to HTML where designs demanded a custom select menu. There is a simple solution to this by W3Schools. You can copy and modify what’s needed without getting deep down to the basics and mechanism.
DEMO
See the Pen Custom select menu by Puneet Sharma (@webdevpuneet) on CodePen.
Credits: https://www.w3schools.com/howto/howto_custom_select.asp