Consider a $p$-adic field $K$ with the standard topology inherited from the usual $p$-adic norm $\mid \cdot \mid$. Consider the ultrametric space $X=K^n$ with the topology inherited from the norm $\| \cdot \|$ defined as $\|x\|=\max_{i=1}^n (|x_1|,\dots,|x_n|)$ with $x=(x_1,\dots,x_n) \in K^n$. Now we have two questions:

Is the topology on $X$ the same as the product topology of the $K$'s?

Can we partition every open compact set in $X$ into a finite number of balls (defined using $\|\cdot \|$)?