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*} \left|\,A|A|^{p-1}-B|B|^{p-1}\,\right|^{2} \leq
|A|^{p-1}\left(\,r|A-B|^{2}+s\left|\,|A|^{1-p}|B|^{p}-|B|\,\right|^2\,\right)|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
¶ $$\left|\,\left(U|A|^{p}-V|B|^{p}\right)|A|^{1-p}\,\right|^{2}\leq
\left(1+t\strut\right)|A-B|^{2}+\left(1+\frac{1}{t}\right) \left| |B|^{p}|A|^{1-p}-|B|
\right|^2 .$$
¶ We obtain several equivalent conditions, when the case of equalities hold.