In this paper, the authors establish the conditions for the extinction of solutions, in finite time, of the fast diffusive p-Laplace equation $u_t=div({mid}{ abla}u{mid}^{p-2}{ abla}u)+a{int}_{Omega}u^q(y,t)dy$, 1 < p < 2, in a bounded domain ${Omega}{subset}R^N$ with $N{geq}1$. More precisely, it is shown that if q > p-1, any solution vanishes in finite time when the initial datum or the coefficient a or the Lebesgue measure of the domain is small, and if 0 < q < p-1, there exists a solution which is positive in ${Omega}$ for all t > 0. For the critical case q = p-1, whether the solutions vanish in finite time or not depends crucially on the value of $a{mu}$, where ${mu}{int}_{Omega}{phi}^{p-1}(x)dx$ and ${phi}$ is the unique positive solution of the elliptic problem -div(${mid}{ abla}{phi}{mid}^{p-2}{ abla}{phi}$) = 1, $x{in}{Omega}$; ${phi}(x)$=0, $x{in}{partial}{Omega}$. This is a main difference between equations with local and nonlocal sources.