Hi,

You have to inherit the payment template of website payment page(website_sale.payment) for adding a new field to it.

You can refer the following example for adding a selection field in payment page of website

<template id="website_payment_inherit" inherit_id="website_sale.payment">
    <xpath expr="//div[@id='payment_method']" position="before">
        <t t-set="field_name_val"
           t-value="request.env['model.name'].sudo().
           search([])"/>
        <select class='detail needsclick' name='field_name'>
            <option value=''>None</option>
            <t t-foreach='field_name_val' t-as='val'>
                <option t-att-value='val.id'>
                    <t t-esc='val.name'/>
                </option>
            </t>
        </select>
    </xpath>
</template>

To know more about Odoo website-controllers visit the following blog

https://www.cybrosys.com/blog/web-controllers-in-odoo

If you new to Odoo development then you can refer the below resources for the future development purpose

https://www.youtube.com/playlist?list=PLeJtXzTubzj-tbQ94heWeQFB0twGd0vvN

Regards