  We provide a partial classification of the 3-strand pretzel knots $K =
P(p,q,r)$ with unknotting number one. Following the classification by Kobayashi
and Scharlemann-Thompson for all parameters odd, we treat the remaining
families with $r$ even. We discover that there are only four possible
subfamilies which may satisfy $u(K) = 1$. These families are determined by the
sum $p+q$ and their signature, and we resolve the problem in two of these
cases. Ingredients in our proofs include Donaldson's diagonalisation theorem
(as applied by Greene), Nakanishi's unknotting bounds from the Alexander
module, and the correction terms introduced by Ozsv\'ath and Szab\'o. Based on
our results and the fact that the 2-bridge knots with unknotting number one are
already classified, we conjecture that the only 3-strand pretzel knots
$P(p,q,r)$ with unknotting number one that are not 2-bridge knots are
$P(3,-3,2)$ and its reflection.
