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We prove that if $f$ is a locally Lipschitz function of exponent $\\alpha \\in (0,1)$ with compact support in $\\mathbb H^n$, then, for a suitable constant $A_{n,p}>0$, $$ \\lim_{s\\rightarrow 0^+}(-\\Delta_{\\mathbb H ^n})_p^sf(x)=A_{n,p}|f(x)|^{p-2}f(x),\\quad x\\in \\mathbb H^n, $$ where $(-\\Delta_{\\mathbb H ^n})_p^s$ denotes the $s$-fractional $p$-Laplacian on $\\mathbb H^n$. 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