We study the field dependence of the maximum supercurrent in narrow edge-type thin-film Josephson junctions. It is assumed that the junction extends across thin-film strip of width W that is much less than the Pearl length; the film thickness is much less than the London penetration depth. We calculate the maximum supercurrent within nonlocal Josephson electrodynamics, which takes into account the stray fields affecting tunneling currents. In the case when W is much less than the thin-film Josephson length, the phase difference along the junction depends only on the junction geometry and the applied field, but is independent of the Josephson critical current density, i.e., it is universal. Zeros of the maximum supercurrent are equidistant only in large fields (unlike the case of junctions with bulk banks); they are spaced by a field that is much smaller than the one of bulk junctions. Peaks of the maximum supercurrent decrease inversely proportional to the square root of the applied field, i.e., slower than 1/H for the bulk.
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