Thank you all for the replies and support.
I followed your suggestions and went on YouTube, picked one of the many videos on how to take apart the Surface Pro 4 to replace the battery.
I used a hair dryer for the glue, and a lot of patience ! I did not break anything, and I just ordered a battery online and by next week I'll have my Surface Pro 4 back.
It is a shame that the policy is that a perfectly functioning computer costing over 2 thousands euros has to be trashed because of a battery costing Microsoft a few dollars.
Environment sustainability anyone? Let's start from making the batteries swappable, as in the past!
Take care all, and thanks again.