I think a counterexample can be obtained as follows. Let $T_1,T_2,\dots$ be independent geometrically distributed random variables with fast growing means, say with $ET_i=2^i$. 

Let $Y_n=0$ for $0\le n\le T_1$. For time moments $n$ after that, let the values of $Y_n$ coincide with the positions of a simple random walk $W^{(1)}_\cdot$ starting from $0$ at time $T_1$ -- but only till the time, say $\nu_1$, of the first return of $W^{(1)}_\cdot$ to $0$. This is the first "step". 

After this, let $Y_\cdot$ stay at $0$ for time $T_2$. After that, let the values of $Y_n$ coincide with the positions of a simple random walk $W^{(2)}_\cdot$ starting in state $0$ at time $\nu_1+T_2$ -- but only till the time, say $\nu_2$, of the first return of $W^{(2)}_\cdot$ to $0$, where $W^{(2)}_\cdot$ is independent of $W^{(1)}_\cdot$. This is the second "step". 

Continue doing such "steps" indefinitely (assuming that the walks are independent of the $T_j$'s). Thus, we get a martingale $(Y_n)$. The walks occur increasingly rarely; they start at very uncertain times (with standard deviations $\asymp2^i$); and they last comparatively short times. Therefore, at any given large time moment $n$, $P(Y_n\ne0)$ is small. So, $Y_n\to0$ in probability. 

However, because the walks occur infinitely many times, clearly $Y_n\not\to0$ almost surely. 

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I will try to formalize this idea later.