JSFiddle - React, Tailwind, and code Playground

HTML

<div>
<select><option>Foobar</option</select>
</div>

CSS

div {
    position:relative;
    display:inline-block;
    border:solid black 1px;
    z-index:0
}
div select {
    z-index:1;
}

div:before {
    display:block;
    position:absolute;
    content:'';
    right:0px;
    top:0px;
    height:1em;
    width:1em;
    margin:2px;
    background:red;
    z-index:5;
}

div:hover::before {
  display: none;
}