The situation is more subtle than suggested by the other two answers as the following example shows.

In $d\ge 2$ dimensions, consider the Euclidean Gaussian field with propagator given in momentum space by $$ \frac{1}{p^{d-2\Delta}} $$ where $\Delta$ is in the interval $\left(\frac{d-2}{2},\frac{d}{2}\right)$. This satisfies the unitary bound and in fact all the Osterwalder-Schrader axioms. Therefore, by analytic continuation to Minkowski space, this results in a QFT that satisfies all the Gårding-Wightman axioms including locality: $$ [\phi(x),\phi(y)]=0 $$ if $x-y$ is space-like.

On the other hand, the Lagrangian for this model is nonlocal.

The situation is more subtle than suggested by the other two answers as the following example shows.

In $d\ge 2$ dimensions, consider the Euclidean Gaussian field with propagator given in momentum space by $$ \frac{1}{p^{d-2\Delta}} $$ where $\Delta$ is in the interval $\left(\frac{d-2}{2},\frac{d}{2}\right)$. This satisfies the unitarity bound and in fact all the Osterwalder-Schrader axioms. Therefore, by analytic continuation to Minkowski space, this results in a QFT that satisfies all the Gårding-Wightman axioms including locality: $$ [\phi(x),\phi(y)]=0 $$ if $x-y$ is space-like.

On the other hand, the Lagrangian for this model is nonlocal.