Will all 21 million units be completely mined one day? It is written that the supply of Bitcoin units is limited to 21 million and that this limit will approximately have been reached around the year 2030.

But is there an actual "last Bitcoin" (or "last Satoshi" for that matter) that will have been mined one day? Or will mining continue forever after producing smaller and smaller fractions?
Theoretically yes, but that moment is so far in the future it is likely that some other events such as new currency technology will surpass bitcoins.

However, mining will not stop.  Mining is the security backbone of bitcoin, and it will continue because it will still be profitable to mine.  Miners will be paid fees from existing bitcoins with every transaction instead of a reward of newly created bitcoins.