After retirement, I decided to label myself as an independent researcher. It implies I am open for contract work, and I am independent of my alma mater. Of course as a product of that university, I still think their way. Anybody can look up where I graduated.
Mentioning the name of your previous institute will require permission from them. Your institute must allow you to use their name, using their name without permission may be illegal. If they allow, well and good, you will not only get the affiliation, but will also be able to get an institutional email ID (.edu, .gov and so on). Without an institutional affiliation you can mention your country's name and city's name along with the pin code. Make sure you also provide your email ID, in case if any related researcher wants to contact you.
Instead, I created an non-profit Institution .Several expenses are tax deductible in my country (México).Only two souls are required for it....My Wife and I .
The problem with simply using "independent researcher" is that people without any academic credentials whatsoever can still call themselves independent researchers. Retirees who are emeriti can of course use their university affiliation; other academic retirees might be given status as adjuncts or affiliates, which may not require active involvement, but allows the university's name to be used. Mere alumni can still indicate their postgraduate degree after their name and granting institution in parentheses: e.g. John Smith, PhD (Oxford)