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Choice A is correct. A line that's tangent to a circle is perpendicular to the radius of the circle at the point of tangency. It's given that the circle has its center at $\left(-4,-6\right)$ and line $k$ is tangent to the circle at the point $\left(-7,-7\right)$. The slope of a radius defined by the points $\left(q,r\right)$ and $\left(s,t\right)$ can be calculated as $\frac{t-r}{s-q}$. The points $\left(-7,-7\right)$ and $\left(-4,-6\right)$ define the radius of the circle at the point of tangency. Therefore, the slope of this radius can be calculated as $\frac{\left(-6\right)-\left(-7\right)}{\left(-4\right)-\left(-7\right)}$, or $\frac{1}{3}$. If a line and a radius are perpendicular, the slope of the line must be the negative reciprocal of the slope of the radius. The negative reciprocal of $\frac{1}{3}$ is $-3$. Thus, the slope of line $k$ is $-3$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the slope of the radius of the circle at the point of tangency, not the slope of line $k$.

Choice D is incorrect and may result from conceptual or calculation errors.