So my question is: is this formula, that I frequently encounter in syllabi and books, correct? Where lies my mistake.
The procedure
$$\mathbf F_{ext} = \frac{d\mathbf p}{dt} = \frac{d}{dt}(m\mathbf v) = \frac{dm}{dt}\mathbf v + m\mathbf a,$$
is based on the erroneous idea that the equation
$$
\mathbf F_{ext} = \frac{d\mathbf p}{dt}
$$
is valid for systems with variable mass ($\mathbf F_{ext}$ is external force on the system). It keeps around probably also because of some relativity texts saying $\mathbf F_{ext} = d\mathbf p/dt$ is more general than $\mathbf F_{ext} = m\mathbf a$ because the former is valid in relativity.
But in non-relativistic mechanics this equation is valid only when the system does not lose or gain parts.
In the more rare case where the system of interest (a control volume = imaginary , possibly moving region in space) does acquire or lose material parts (like a rocket), the equation
$$
\mathbf F_{ext} = \frac{d\mathbf p}{dt}
$$
is no longer valid. Another equation may be derived from this one applied to the whole supersystem (system + incoming/leaving parts). This can be done because the supersystem does not lose or gain particles. The new equation is
$$
\mathbf F_{ext} + \mathbf F_{parts} = \sum_k m_k \mathbf a_k
$$
where $\mathbf F_{parts}$ is force on the system due to parts no longer inside the control volume and summation is to be done over all particles in the control volume.
It can be written also in this way:
$$
\mathbf F_{ext} + \mathbf F_{parts} = \frac{d\mathbf p}{dt} + \frac{d\mathbf p_{lost}}{dt}
$$
where $d\mathbf p_{lost}$ is momentum lost from the control volume per time $dt$.
In the simplest case, where the lost particles all leave in direction same or opposite to the body velocity $\mathbf v$ (idealized rocket), this can be further simplified. Let the boundary of the control volume be far from the rocket, so that velocity of particles crossing the boundary (relatively to the rocket) is constant $\mathbf{c}$, and $\mathbf{F}_{parts}$ is negligible. Then the lost momentum per unit time is
$$
\frac{d\mathbf{p}_{lost}}{dt} = - \frac{dm}{dt}(\mathbf v+\mathbf c)
$$
and the equation of motion simplifies into
$$
\mathbf{F}_{ext} + \frac{dm}{dt} \mathbf c = m\frac{d\mathbf{v}}{dt}.
$$
It is important to realize also that $\mathbf v$ is not velocity of center of mass of the system, but is defined as $\mathbf p/m$ where $\mathbf p$ and $m$ are momentum and mass inside the control volume. The necessity of this distinction is best seen from this example: let the body have constant velocity $\mathbf v$, but let the control volume shrink so that less and less of the body is inside. Center of mass of the control volume has different velocity from $\mathbf v$, in fact it accelerates due to moving boundary of the control volume. However, velocity of the material particles does not change at all.
