I claim that for any infinite field $K$, there is such a linearly independent almost-disjoint family $X$ 
of size at least continuum. This is optimal when the field $K$ has size at most continuum, since in this case $K^{\mathbb{N}}$ has size continuum. 

Proof: We shall build a finitely branching tree $T\subset K^{\lt\mathbb{N}}=\bigcup_n K^n$ with the following properties: 

 - the $n^{th}$ level of the tree consists of a linearly independent collection in $K^n$. 
 - the tree is splitting, in the sense that every node in $T$ has incomparable extensions in $T$. 
 - distinct sequences in $T$ have no agreement beyond their common initial segment. In other words, once two sequences disagree, they never agree again. This is equivalent to insisting that the values of $K$ arising on a given level of the tree are distinct.

Given such a tree, consider the collection $X$ of all paths through the tree. This will have size continuum, since the tree is splitting. Distinct paths through $T$ will have only finite agreement, since they disagree beyond their common initial segment, and so $X$ is an almost disjoint family. And any finitely many branches from $X$ will be linearly independent, since they will be linearly independent even when restricted to any level of the tree where those branches become distinct. 

So let's build the tree. Suppose that it has been specified up to level $n$, consisting of $m$ linearly independent sequences in $K^n$. Extend each of these sequences by appending a distinct non-zero element of $K$ to it, choosing one of the nodes to receive two such extensions. I claim that the resulting family is still linearly independent in $K^{n+1}$. 

Now, the point is that since the tree will have only countably many nodes, we can arrange to choose the splitting nodes in such a way that every node leads eventually to a splitting node. (e.g. we could handle the finitely many nodes at level $n$ in the tree one after the other.)

So we've built the tree as desired, and so there is a linearly independent almost disjoint family of size continuum. QED