I have a theory, which is basically that 'most pumps are junk'. There are millions of different ones and nearly all of them are deeply flawed in either conception or execution.
Like politicians, you choose 'the least bad option available' and there is a small chance that, once selected, something vaguely useful might happen, for a while at least. And then it all goes belly up.
Think about it; with few exceptions, I'd say that most cyclists have had more pumps than bicycles, even though the pump might only get a few minute's use in a year!
So what to do.... I think that Meic's advice is sound. I'd also add that it makes sense to maintain your pump, using some kind of lube that isn't going to dry out or corrupt the seals.
For lube, don't
use 'oil' unless you are in dire need; most oils corrupt various different rubbers, which can include the pump workings, but will include your inner tubes. For a cheap lube that seems to work OK in pumps (doesn't attack plastics and rubber, doesn't dry out appreciably) I'd suggest '151 super grease'.
I'd also add that you shouldn't expect miracles; obviously a tiny pump will take much longer to inflate a tyre, and 'clever designs' (ie dual action, telescoping, etc etc) IME just add places where the pump is less efficient and more likely to go wrong; even if they do 'work' they may be no faster or easier than a pump with a simpler design.
Also, many pumps (not just mini-pumps) use an 'O' ring as a main pump seal. This won't ever last as long as a proper cup washer, even if lubricated properly, but if it has run dry, I'd expect it to fail entirely within the length of time that it takes to inflate one or two tyres, which is probably what has happened to Drake's pump.