= Solution
Continuity on the compact interval ensures that a maximum time exists. Fix a rational $r\in(0,1)$ and define
$$
U_r=\max_{0\leq t\leq r}B_t-B_r,
\qquad
V_r=\max_{r\leq t\leq1}(B_t-B_r).
$$
By Brownian time reversal and the <Brownian running maximum> law, $U_r\stackrel d=|N(0,r)|$. Independent increments give $V_r\stackrel d=|N(0,1-r)|$ independently of $U_r$. Their continuous distributions imply $\mathbb P(U_r=V_r)=0$.
If the maximum were attained at two distinct times, a rational $r$ strictly between them would make the maxima on $[0,r]$ and $[r,1]$ equal, hence $U_r=V_r$. A countable union over rational $r$ still has probability zero. Therefore the <time of the Brownian maximum> is almost surely unique.
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