Broken Apple TV remote 1st generation

Apple only sells the latest model at this time, which is compatible with Apple TV HD (4th generation) [A1625] or later. It would likely be a suitable replacement for the black Siri Remote (unless you need the gyro and accelerometer for e.g. games). This current model remote requires tvOS 15.4 or later.

Identify your Apple TV model - Apple Support

Find your Apple TV model number - Apple Support

Apple TV remote models and their compatibility - Apple Support

… needs battery replacement as well as a clean as there are lots of dust

Battery service means whole remote replacement. Only the legacy remotes had a replaceable battery.

There is no official cleaning service, but you may find someone willing to do some wiping for you in exchange of cash.

if it's black one then it's not the first generation one

this is the 1. 2. and 3. generation appletv remotes

this 4. one is the only one they currently sell

it should work with older models

otherwise you will need to look for third party options

Amazon.com : appletv remotes