= Solution
Let $U_t=(1,0,0)+(B_t^1,B_t^2,B_t^3)$, with three <independent> standard <Brownian motions>, and set $R_t=|U_t|$ and $M_t=R_t^{-1}$. The radial <stochastic process> $R$ is a three-dimensional <Bessel process>, which never hits zero by the <Hitting-zero classification for a Bessel process>. In three dimensions $u\mapsto|u|^{-1}$ is a <harmonic function> away from zero: direct differentiation gives $\Delta(|u|^{-1})=0$.
For the <stopping times> $T_n=\inf\{t:R_t\notin(1/n,n)\}$, $n\geq2$, the <Itô formula> gives
$$
M_{t\wedge T_n}=1-\sum_{j=1}^3\int_0^{t\wedge T_n}\frac{U_s^j}{R_s^3}\,dB_s^j.
$$
The coefficients are bounded by $n^2$, so each stopped <Itô integral> is a true <martingale> on every finite horizon. The <stopping times> form a <localizing sequence>; as permitted, its divergence need not be proved here. Thus $M$ is a nonnegative <continuous local martingale>.
To see that $M$ is not a true <martingale>, integrate the three-dimensional <multivariate normal density> of $U_t$ in <spherical coordinates>. The angular integral yields
$$
\begin{aligned}
\mathbb EM_t
&=\frac1{\sqrt{2\pi t}}\int_0^\infty\left(e^{-(r-1)^2/(2t)}-e^{-(r+1)^2/(2t)}\right)dr\\
&=2\Phi(1/\sqrt t)-1<1=\mathbb EM_0\qquad(t>0),
\end{aligned}
$$
where $\Phi$ is the <standard normal distribution function>. Therefore \b[the reciprocal of a three-dimensional <Bessel process> is a <strict local martingale>.]
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