Can the number of points at integral distance to all three points of a non-degenerate triangle of area $A$ be bounded by $1+cA$ for some suitable constant $c$?

Remark: Since it is easy to bound this number by $4(D+1)^2$ where $D$ is the diameter of the triangle, a sequence giving rise to counterexamples must consist of very thin triangles.

Addendum: Ilya Bogdanov's example shows that the original statement is too optimistic. There are two ways (both suggested in Ilya's answer) to strengthen it: Replace $1$ by a constant or ask the question asymptotically (for $A$ large enough). Both strengthenings are interesting but I prefer the first one:

Are there constants $b\geq 2$ and $c$ such that the number of points at integral distance to a triangle of area $A$ is bounded by the affine function $b+cA$ of the area?