  We will give a criterion to assure that an extremal contraction of a K3
surface which is not a Mori Dream Space produces a singular surface which is a
Mori Dream Spaces. We list the possible N\'eron--Severi groups of K3 surfaces
with this property and an extra geometric condition such that the Picard number
is greater then or equal to 10. We give a detailed description of two geometric
examples for which the Picard number of the K3 surface is 3, i.e. the minimal
possible in order to have the required property. Moreover we observe that there
are infinitely many examples of K3 surfaces with the required property and
Picard number equal to 3.
