The correct order of Lewis acid strength of boron trihalides towards a given base (such as $\mathrm{NH}{}_3$) is

Back donation of the halogen lone pair into the empty $2p$ orbital of boron is $p\pi$-$p\pi$ overlap. It is strongest for $\mathrm{F}$ (same period, similar size), and falls on moving to Cl and Br. Less back donation leaves boron more electron deficient, so the acid strength is $\mathrm{BBr}{}_3>\mathrm{BCl}{}_3>\mathrm{BF}{}_3$.