JSFiddle - React, Tailwind, and code Playground

HTML

<div id="chatApplicationClassic">
  <div id="chatHeaderClassic">
    <div id="chatHeaderArrowClassic">
      Arr
    </div>
    <div id="chatHeaderTextClassic">
      Top
    </div>
  </div>
  <div id="chatBodyClassic">
    bod
    <br /> bod
    <br /> bod
    <br /> bod
    <br /> bod
    <br /> bod
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    <br /> bod
    <br /> bod
    <br /> bod
    <br /> bod
    <br /> bod
    <br /> bod
    <br /> bod
    <br /> bod
    <br />
  </div>
  <!--
	<div id="chatFooterClassic">
		<div id="chatInputContainerClassic">
			<input placeholder="type here..." id="chatInputClassic" type="text" /> 
		</div>
		<div id="chatSendButton">
			<div id="chatSendButtonText">Send</div>
		</div>
	</div>-->
</div>

CSS

#chatApplicationClassic {
  padding: 0;
  margin: 0;
  border: 0;
  position: fixed;
  bottom: 0px;
  right: 30px;
  width: 300px;
  background-color: lime;
  box-sizing: border-box;
  border-radius: 5px 5px 0 0;
  box-shadow: 0px 0px 5px #000000;
}

#chatHeaderClassic {
  padding: 0;
  margin: 0;
  border: 0;
  width: 100%;
  background-color: pink;
  cursor: pointer;
  border-radius: 5px 5px 0 0;
  border-top: 0px solid red;
  border-left: 0px solid red;
  border-right: 0px solid red;
  border-bottom: 0px solid red;
  box-sizing: border-box;
  overflow: hidden; /*added*/
}

#chatHeaderTextClassic {
  padding: 0;
  margin: 0;
  border: 0;
  color: #ffff33;
  font-family: 'Open Sans', sans-serif;
  padding: 6px;
  font-size: 18px;
  font-weight: 100;
  /* display: inline-block; */
  border-radius: 5px 5px 0 0;
  float: left; /*added*/
}

#chatHeaderArrowClassic {
  padding: 0;
  margin: 0;
  border: 0;
  color: #9900ff;
  /* display: inline-block; */
  float: right;
  padding: 6px;
  font-size: 18px;
}

#chatBodyClassic {
  padding: 0;
  margin: 0;
  border: 0;
  background-color: white;
  border-left: 5px solid pink;
  border-right: 5px solid pink;
  height: 200px;
  width: 100%;
  box-sizing: border-box;
  padding-top: 10px;
  overflow: scroll;
}

.chatAgentBubbleClassic {
  padding: 0;
  margin: 0;
  border: 0;
  background-color: #664eff;
  margin-right: 40px;
  margin-top: 10px;
  box-sizing: border-box;
  border-radius: 4px;
}

.chatVisitorBubbleClassic {
  padding: 0;
  margin: 0;
  border: 0;
  background-color: #6600ff;
  margin-left: 40px;
  margin-top: 10px;
  box-sizing: border-box;
  border-radius: 4px;
}

.chatAgentBubbleTextClassic {
  font-size: 12px;
  padding: 8px;
  font-family: sans-serif;
  color: lightblue;
}

.chatVisitorBubbleTextClassic {
  font-size: 12px;
  padding: 6px;
  font-family: sans-serif;
  color: skyblue;
}