Z-offset issues with Slic3r and Da Vinci Nano
Behavioral analysis
To find out what is actually wrong, let's do a simple series of test commands first: a G-code that homes the printhead, moves it up 100 mm (10 cm), then zeroes there, moves up 20 more and back to the new 0 at 100 mm above the bed. Make sure not a single offset is active anywhere when sending/loading the G-code as this is to test for the actual movement.
G28 
G1 Z100 
G92 Z0 
G90 
G1 Z20 
G1 Z0

Measuring the distance to the bed should give us something hopefully close to 100 mm. If it is longer or shorter, we have a problem with the bed position to the endstop, even if the bed itself is level to the machine. A releveling is advised unless there is a reason to home to a position widely outside of the printing height or if leveling proves to be impossible.
Why is the error experienced?!
The culprit seems to be that the Z-offset parameter in slic3r is not updated properly when altering it. I just tested this with a ludicrous offset and it took several saving and slicing attempts to register that Z0 was supposed to be 200 mm over the endstop via the offset setting. Compare these two blocks:
G90 ; use absolute coordinates
M82 ; use absolute distances for extrusion
G92 E0
G1 Z0.350 F7800.000

And this one:
G90 ; use absolute coordinates
M82 ; use absolute distances for extrusion
G92 E0
G1 Z-199.650 F7800.000

The offset only appears factored in starting with the actual print's G1 commands. But how did I eventually manage to make it update?
Forcing the update
To force an update of all the settings, I did create a secondary printer profile (just saved the same one and in the new one set the offset back to 0) and then swapped between the two back and forth twice and then the correct offset one. This seems to be a very reliable way to force an update of this setting, as it did register every swap after the first one.
Workarounds
manual Offsetting via Filament parameter
But there is a way around that is always taken into account: we can add a custom Filament Start Code, which contains the needed offset under Filament Settings tab &gt; Custom G-Code:
G1 Z&lt;offset we need to be exactly at layer 0 height&gt;
G92 Z0.3

But this is a rather clumsy way and should only be made if nothing else helps and not in conjunction with any offset in the machine setting!
Tackle the problem by the root
The problem isn't usually the offset but a bed leveled to the wrong height. To level the bed the printer should be brought to Z0 via G28 and then leveled again. If this proves futile, setting the offset and forcing an update in slic3r is needed.