<div class="datepicker-desktop">
    <div class="inline-details">
        <noscript>
            Enter a date in the following format YYYY-MM-DD.
        </noscript>
    </div>

    <input type="hidden" name="{$Name}_formatted" >
    <input $AttributesHTML data-date-inputformat="D/M/YYYY"
           data-date-displayformat="DD/MM/YYYY"
           placeholder="for example, 27/3/1972" />

    <%-- if $RightTitle %>aria-describedby="{$Name}_right_title" <% end_if --%>

    <button class="accCalendar datePicker datepicker-button" aria-describedby="{$Id}" aria-expanded="false"
        aria-label="You can use this calendar to pick a date">
        <img src="$resourceURL(govtnz/silverstripe-userforms-datepicker:client/javascript/bootstrap-jquery/img/calendar/calendar-button.svg)"
             alt="You can use this calendar to pick a date"
             title="You can use this calendar to pick a date">
    </button>
    <div class="clear"></div>
</div>

<div class="datepicker-desktop">
    <div class="inline-details">
        <noscript>
            Enter a date in the following format YYYY-MM-DD.
        </noscript>
    </div>

    <input type="hidden" name="{$Name}_formatted" >
    <input $AttributesHTML data-date-inputformat="D/M/YYYY"
           data-date-displayformat="DD/MM/YYYY"
           placeholder="for example, 27/3/1972" />

    <%-- if $RightTitle %>aria-describedby="{$Name}_right_title" <% end_if --%>

    <button type="button" class="accCalendar datePicker datepicker-button" aria-describedby="{$Id}" aria-expanded="false"
        aria-label="You can use this calendar to pick a date">
        <img src="$resourceURL(govtnz/silverstripe-userforms-datepicker:client/javascript/bootstrap-jquery/img/calendar/calendar-button.svg)"
             alt="You can use this calendar to pick a date"
             title="You can use this calendar to pick a date">
    </button>
    <div class="clear"></div>
</div>

Add type="button" to calendar button
Add type="button" to calendar button

So it doesn't default type="submit" and submit  the form when clicked. 