= Solution
Localize to compact subintervals of $(0,\infty)$, so that the <Itô formula> can be applied to $f(z)=\sqrt z$. Its derivatives are $f'(z)=1/(2\sqrt z)$ and $f''(z)=-1/(4z^{3/2})$, while $d[Z]_t=4Z_tdt$. Thus, before $\zeta$,
$$
\begin{aligned}
dR_t&=\frac1{2R_t}(2R_t\,dB_t+\gamma dt)
-\frac1{8R_t^3}(4R_t^2dt)\\
&=dB_t+\frac{\gamma-1}{2R_t}dt.
\end{aligned}
$$
Hence the <Bessel process> equation is
$$
\boxed{dR_t=dB_t+\frac{\gamma-1}{2R_t}dt,\qquad R_0=r,\quad t<\zeta.}
$$
The inverse-power drift is interpreted only while $R>0$; its boundary behaviour is justified next, not assumed in this calculation.
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