  We present several operator versions of the Dunkl--Williams inequality with
respect to the $p$-angular distance for operators. More precisely, we show that
if $A, B \in \mathbb{B}(\mathscr{H})$ such that $|A|$ and $|B|$ are invertible,
$\frac{1}{r}+\frac{1}{s}=1\,\,(r>1)$ and $p\in\mathbb{R}$, then
\begin{equation*} |A|A|^{p-1}-B|B|^{p-1}|^{2} \leq
|A|^{p-1}(r|A-B|^{2}+s||A|^{1-p}|B|^{p}-|B||^2)|A|^{p-1}.%\nonumber
\end{equation*} In the case that $0<p \leq 1$, we remove the invertibility
assumption and show that if $A=U|A|$ and $B=V|B|$ are the polar decompositions
of $A$ and $B$, respectively, $t>0$, then
$$|(U|A|^{p}-V|B|^{p})|A|^{1-p}|^{2}\leq
(1+t)|A-B|^{2}+(1+\frac{1}{t})||B|^{p}|A|^{1-p}-|B||^2 \,.$$ We obtain several
equivalent conditions, when the case of equalities hold.
