Wow, thanks everyone for your suggestions on drill bits ! 
I will write it all down for future reference... looks like this problem is solved now. 
Hi Vince,
As usual, having the right tool is just halfway of the achivement. For happy drilling, there are some good practices to follow.
For every matter (or at least families of matters), dedicated drill bits are available.
The harder a drill bit, the more the cutting lips are brittle and won't survive any serious jolt.
Don't let any Tungsten-carbide drill bit fall down on a hard surface (concrete) : il will multiply immediately.
Heat is a deadly enemy for drill bits. As normal steel starts to temper at 200°C, HSS (High Speed Steel) withstands up to
650°C, best case. An overheated drill bit is to be considered as dead and can be thrown in the bin.
Using tooling machine lubricant (cutting fluid) significantely increases the lifespan of your drill bits.
Best is to setup a
lubricant pump (*) and some
Locline flex pipes.
Set the speed of your drill press according to the drill bit manufacturer recommendations and the matter you're machining.
Using a vanilla drill bit on sheet metal is a bad idea : prefer the hole saw or a staged drill bit.
Below 3 mm, drill bits are to be considered as expendable and cannot be sharpened.
For uncoated drill bits, using a
dedicated sharpener (here
another one) is a worthwhile investment, although results are often average. But it's better than nothing.
(*) Same gear can be cobbled together for less that 100 bucks by using a
paint bucket and a
suitable pump.
Cutting fluid / lubricant (mix 3-8% with plain water) :
https://befr.rs-online.com/web/p/cutting-fluids/1316306https://www.amazon.fr/DLLUB-SOLUBLE-DUSINAGE-2000-5-litres/dp/B00QFJK16Shttps://www.amazon.fr/DLLUB-SOLUBLE-DUSINAGE-1000-5-litres/dp/B00QFJK8E8https://xometry.eu/en/cutting-fluids-for-cnc-machining-processes/