We prove an analogue for $p$-adic coefficients of the Deligne--Laumon theorem on local acyclicity curves. That is, overconvergent $F$-isocrystal $E$ a relative curve $f:U\rightarrow S$ admitting good compactification, we show that cohomology sheaves $\mathbf{R}f_!E$ are isocrystals if and only has constant Swan conductor at infinity.