1. Given: Four particles have the same momentum $p$.

   Their masses are:

   • Particle $A: \dfrac{m}{2}$
   • Particle $B: m$
   • Particle $C: 2m$
   • Particle $D: 4m$

2. Kinetic energy in terms of momentum:

   $K = \frac{p^2}{2M}$

   where $M$ is the mass of the particle.

3. Since all particles have the same momentum $p$, kinetic energy is:

   $K \propto \frac{1}{M}$

   So, smaller the mass, larger the kinetic energy.

4. Compare the masses:

   $\frac{m}{2},\; m,\; 2m,\; 4m$

   The smallest mass is $\dfrac{m}{2}$, which belongs to particle $A$.

5. Therefore, particle $A$ has the maximum kinetic energy.

6. Check options:

   • Option A: $B$ ❌
   • Option B: $C$ ❌
   • Option C: $D$ ❌
   • Option D: $A$ ✅

Hence, the correct answer is Option D.