I had this exact problem, except I couldn't use images and was not limited by browser support. This should be «on spec» and with luck start working everywhere eventually.
It uses layered rotated background layers to «cut out» a dropdown arrow, as pseudo-elements wouldn't work for the select element.
Edit: In this updated version I am using CSS variables and a tiny theming system.
:root {
--radius: 2px;
--baseFg: dimgray;
--baseBg: white;
--accentFg: #006fc2;
--accentBg: #bae1ff;
}
.theme-pink {
--radius: 2em;
--baseFg: #c70062;
--baseBg: #ffe3f1;
--accentFg: #c70062;
--accentBg: #ffaad4;
}
.theme-construction {
--radius: 0;
--baseFg: white;
--baseBg: black;
--accentFg: black;
--accentBg: orange;
}
select {
font: 400 12px/1.3 sans-serif;
-webkit-appearance: none;
appearance: none;
color: var(--baseFg);
border: 1px solid var(--baseFg);
line-height: 1;
outline: 0;
padding: 0.65em 2.5em 0.55em 0.75em;
border-radius: var(--radius);
background-color: var(--baseBg);
background-image: linear-gradient(var(--baseFg), var(--baseFg)),
linear-gradient(-135deg, transparent 50%, var(--accentBg) 50%),
linear-gradient(-225deg, transparent 50%, var(--accentBg) 50%),
linear-gradient(var(--accentBg) 42%, var(--accentFg) 42%);
background-repeat: no-repeat, no-repeat, no-repeat, no-repeat;
background-size: 1px 100%, 20px 22px, 20px 22px, 20px 100%;
background-position: right 20px center, right bottom, right bottom, right bottom;
}
select:hover {
background-image: linear-gradient(var(--accentFg), var(--accentFg)),
linear-gradient(-135deg, transparent 50%, var(--accentFg) 50%),
linear-gradient(-225deg, transparent 50%, var(--accentFg) 50%),
linear-gradient(var(--accentFg) 42%, var(--accentBg) 42%);
}
select:active {
background-image: linear-gradient(var(--accentFg), var(--accentFg)),
linear-gradient(-135deg, transparent 50%, var(--accentFg) 50%),
linear-gradient(-225deg, transparent 50%, var(--accentFg) 50%),
linear-gradient(var(--accentFg) 42%, var(--accentBg) 42%);
color: var(--accentBg);
border-color: var(--accentFg);
background-color: var(--accentFg);
}
<select>
<option>So many options</option>
<option>...</option>
</select>
<select class="theme-pink">
<option>So many options</option>
<option>...</option>
</select>
<select class="theme-construction">
<option>So many options</option>
<option>...</option>
</select>