Hi nauna,

Refer below code.

HTML

<script type="text/javascript" src="https://ajax.googleapis.com/ajax/libs/jquery/1.8.3/jquery.min.js"></script>
<script type="text/javascript">
    function ScrollPanelBottom() {
        var text =  $('.container');
        text.scrollTop(text[0].scrollHeight);
    };

    var prm = Sys.WebForms.PageRequestManager.getInstance();
    if (prm != null) {
        prm.add_endRequest(function (sender, e) {
            if (sender._postBackSettings.panelsToUpdate != null) {
                var text = $('.container');
                text.scrollTop(text[0].scrollHeight);
            }
        });
    };

    $(function () {
        $('.container').on('mousedown', function (e) {
            $('#hfX').val(event.pageX);
        });

        $('.container').on('mouseup', function (e) {
            if (parseInt($('#hfX').val()) >= e.pageX) {
                var timer = $find('<%=timer.ClientID %>');
                timer._stopTimer();
            }
        });

        $('.container').on('mousewheel', function (e) {
            var timer = $find('<%=timer.ClientID %>');
            if (e.originalEvent.wheelDelta >= 0) {
                timer._stopTimer();
            } else {
                timer._startTimer();
            }
        });

        $('.content').on('mousedown', function (e) {
            e.stopPropagation();
        });
    });
</script>
<asp:ScriptManager runat="server" />
<asp:HiddenField ID="hfX" runat="server" />
<asp:Button ID="Button1" runat="server" Text="Button" OnClick="Button1_Click" />
<br />
<div class="comment">
    <asp:Panel ID="Panel2" class="container" runat="server" Height="100px" ScrollBars="Vertical"
        Style="background-color: antiquewhite;">
        <div class="content">
            <asp:UpdatePanel runat="server" ID="upecompetechatlist" UpdateMode="Conditional">
                <ContentTemplate>
                    <asp:Timer ID="timer" runat="server" Interval="1000" OnTick="Timer1_Tick">
                    </asp:Timer>
                    <asp:Label ID="lbltext" runat="server"></asp:Label>
                </ContentTemplate>
                <Triggers>
                    <asp:AsyncPostBackTrigger ControlID="timer" EventName="Tick" />
                </Triggers>
            </asp:UpdatePanel>
        </div>
    </asp:Panel>
</div>

Code

C#

static int a = 1;
protected void Timer1_Tick(object sender, EventArgs e)
{
    // your stuff to refresh after some interval
    lbltext.Text += "<br/> Message " + a;
    a = a + 1;
}

protected void Button1_Click(object sender, EventArgs e)
{
    string script = "window.onload = function() { ScrollPanelBottom(); };";
    ClientScript.RegisterStartupScript(this.GetType(), "message", script, true);
}

VB.Net

Shared a As Integer = 1
Protected Sub Timer1_Tick(ByVal sender As Object, ByVal e As EventArgs)
    lbltext.Text += "<br/> Message " & a
    a = a + 1
End Sub

Protected Sub Button1_Click(ByVal sender As Object, ByVal e As EventArgs)
    Dim script As String = "window.onload = function() { ScrollPanelBottom(); };"
    ClientScript.RegisterStartupScript(Me.GetType(), "message", script, True)
End Sub

Screenshot