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We add a minimal correction term to the local Gross-Pitaevskii equation torepresent non-locality in the interactions. We show that the effective minimalnon-locality can make the healing length decrease more rapidly with theincrease of $s$-wave scattering length leaving the expression of the velocityof sound unaltered. We discuss the implication of this result for aBose-Einstein Condensate (BEC) being used as an analogue gravity system. Thepresented result is important in the context of condensed matter physics aswell because one can considerably change the size of a quantized vortex atfinite $s$-wave scattering length by tuning the healing length.

Nonlocal interactions in a BEC: an Analogue Gravity perspective