Remove Datalist Dropdown Arrow in Chrome

Thanks to the comment by alexander farkas, here is the style rule to remove the arrow:

input::-webkit-calendar-picker-indicator {
  display: none;
}

As others have mentioned ::-webkit-calendar-picker-indicator { display: none } works at removing the arrow it would also impact the html5 datepicker on a <input type="date">,

To remove just removing the datalist input would be:

[list]::-webkit-calendar-picker-indicator { display: none; }

Thanks to Cantera. I didn't want to get rid of the black arrow entirely, just the gray square surrounding it.

input::-webkit-calendar-picker-indicator {
  background-color: inherit;
  }

input::-webkit-calendar-picker-indicator {
  opacity: 0;
}

Also removed the arrow for me and I found created a better clicking experience to still click where the arrow would be, you can even increase the the width and height of it > 1em and in the input maybe put a custom arrow as a background image, where the arrow would be.