This norm is not Gateaux differentiable for $n\ge2$. 

Indeed, let $x:=(2,4,0,\ldots,0)$ and $u:=(1,1,0,\ldots,0)$. Then 
$$g(t):=\|x+tu\|_0=\max(|1+\tfrac t4|,|1+\tfrac t2|) \\
=-(1+\tfrac t2)\,1(t\le-\tfrac83)+(1+\tfrac t4)\,1(\tfrac83<t\le0)
+(1+\tfrac t2)1(t>0)$$
for real $t$.
So, $g$ is not differentiable at $0$ (and also at $-8/3$). $\quad\Box$ 

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Here is the graph $\{(t,g(t))\colon-4<t<1\}$:

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/s6PII.png