  Under ABC, Silverman showed that there are infinitely many non-Wieferich
primes with respect to any (non-trivial) base $a$. Recently Srinivas and
Subramani proved an analogous result over number fields with trivial class
group. In the first part of this article, we extend their result to any
arbitrary number fields. Secondly, we give an asymptotic lower bound for the
number of non-Wieferich prime ideals. Furthermore, we show a lower bound of
same order is achievable for non-Wieferich prime ideals having norm congruent
to $1 \pmod k.$ Lastly, we generalize Silverman's work for elliptic curves over
arbitrary number fields following the treatment by K\"uhn and M\"uller.
