Neutron stars receive velocity kicks at birth in supernovae. Those formed in electron-capture supernovae from superasymptotic giant branch stars - the lowest mass stars to end their lives in supernovae - may receive significantly lower kicks than typical neutron stars. Given that many massive stars are members of wide binaries, this suggests the existence of a population of low-mass (1.25 < M-psr/M-circle dot < 1.3), wide (P-orb greater than or similar to 10(4) d), eccentric (e similar to 0.7), unrecycled (P-spin similar to 1 s) binary pulsars. The formation rate of such binaries is sensitive to the mass range of (effectively) single stars leading to electron capture supernovae, the amount of mass lost prior to the supernova, and the magnitude of any natal kick imparted on the neutron star. We estimate that one such binary pulsar should be observable in the Milky Way for every 10 000 isolated pulsars, assuming that the width of the mass range of single stars leading to electron-capture supernovae is less than or similar to 0.2 M-circle dot, and that neutron stars formed in electron-capture supernovae receive typical kicks less than 10 km s(-1). We have searched the catalogue of observed binary pulsars, but find no convincing candidates that could be formed through this channel, consistent with this low predicted rate. Future observations with the Square Kilometre Array may detect this rare sub-class of binary pulsar and provide strong constraints on the properties of electron-capture supernovae and their progenitors.