ICP-OES data treatment: correct for dilution and inweight, the units
I think you have a typo in your calculation (also, using both dots and commas as the decimal point is rather confusing), but you're on the right track. The unfortunate thing about elemental analysis is that %, ppm, ppb, ppt, etc. get misused in a confusing way. These units are supposed to be used for only dimensionless ratios, the units you start with, ppm is actually mg/L which represents mass concentration, not mass ratio. Your calculations are correct:

$$\left(186.49 \frac{\mathrm{mg}}{\mathrm{L}}\right)\left(0.0142\mathrm{\ L}\right) = 2.648 \mathrm{\ mg}$$
gives the mass of the Mn in your sample, so when you divide by the sample mass:

$$\frac{2.648 \mathrm{\ mg}}{10.17\mathrm{\ mg}}=0.260$$
you get the mass fraction of Mn in your sample, and you know it's a mass fraction because you have mass units in both the numerator and the denominator, leaving you dimensionless. You can express it just like that, or you can convert it to 26.0% (and this is a dimensionless ratio, so it actually makes sense to use % here), or else express it as the mass of Mn in an arbitrary sample mass: 26.0 g/100 g.