Nope, the total momentum $\vec p$ (and similarly the force $\vec F$, which is its time derivative) is just one vector for the whole system (or whole space) so it makes no sense to differentiate it with respect to spatial coordinates, and $\nabla$ is a symbol for the differentiation with respect to $x,y,z$.
That doesn't mean that equations similar to those you are thinking about don't exist. In the mechanics of liquids, solids, and gases, one may talk about the "density" of forces and density of energy and momentum: we want to know not only the total momentum or force but also "where it is located". The density of momentum is combined with the flux of momentum which is expressed by the ($3\times 3$) stress tensor (which is generalized to a $4\times 4$ stress-energy tensor in relativity). There exist equations involving a gradient when one talks about the stress tensor although this page
doesn't really offer too many of them... You may see the conservation law for the stress-energy tensor
which is a generalization of Newton's second law along the lines you proposed. It has gradients but the right hand side is zero: that's because the force is immediately rewritten as the time derivative of the momentum carried by other parts of the physical system.