  Since the three dimensional (3D) Dirac semi-metal Cd$_3$As$_2$ exists close
to topological phase boundaries, in principle it should be possible to drive it
into exotic new phases, like topological superconductors, by breaking certain
symmetries. Here we show that the mesoscopic point-contacts between silver (Ag)
and Cd$_3$As$_2$ exhibit superconductivity up to a critical temperature (onset)
of 6 K while neither Cd$_3$As$_2$ nor Ag are superconductors. A gap amplitude
of 6.5 meV is measured spectroscopically in this phase that varies weakly with
temperature and survives up to a remarkably high temperature of 13 K indicating
the presence of a robust normal-state pseudogap. The observations indicate the
emergence of a new unconventional superconducting phase that exists only in a
quantum mechanically confined region under a point-contact between a Dirac
semi-metal and a normal metal.
