Can a collision between a large and small mass result in final velocity greater than initial velocity?

The best way is to find out, using calculations, Suppose initially mass $m$ is stationary and mass $M$ is moving with speed $u_1$, then if after the elastic collision the velocities are $v_2$ and $v_1$ respectively then $$v_2-v_1=u_1$$ $$Mu_1=Mv_1+mv_2$$ $$v_1=frac{M+m}{M-m}u_1 text{and} v_2=frac{2M}{M-m}u_1$$

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Is it possible for a collision to occur that results in the smaller mass having a velocity $v_f>v$?

$$v_2>u_1Rightarrow frac{2M}{M-m}u_1>u_1Rightarrow 2M>M-mRightarrow M>-m$$ which is true. So it's possible.

all momentum is transferred to the smaller mass? Or will the larger mass always retain some of the momenta?

Suppose the case, In which the $v_1=0$ so that $$M+m=0Rightarrow M=-m$$ which is not possible.