This is definitely false with requiring ample.

Let $M=M_1\times M_2$ and $D_i$ the pull back of an anti-canonical divisor on $M_i$. Then $D_1+D_2\in |-K_M|$ but they are individually not ample. (They are nef though). 

I don't know what you $\lambda$ is, but if you want coefficients strictly between $0$ and $1$, then choose two different anti-canonical divisor on each factor and multiply them by $1/2$. 

To get an example with not nef components take $M$ to be the blow-up of $\mathbb P^n$ in a single point with $E\simeq \mathbb P^{n-1}$ the exceptional divisor. Then $M$ is a $\mathbb P^1$-bundle on $\mathbb P^{n-1}$ and an anti-canonical divisor is something nef pulled-back from the $\mathbb P^{n-1}$ plus a multiple of $E$ which is not nef.

Now the good(?) news is that this is hinges on the fact that $M$ is a blow-up and I believe that for $n>2$, $\mathbb P^n$ is the only manifold whose blow-up is Fano, so you might be able to get some kind of a statement by assuming that $M$ is not rational or something like that.