Our own Messenger satellite put into orbit around Mercury could not maintain a stable orbit due to how close Mercury is to the Sun.
That is an exaggeration!
Mercury's proximity to the Sun did complicate insertion into an orbit around it, but once it was inserted, Messenger remained in orbit for four years.
NASA had deliberately chosen a highly elliptical orbit for Messenger, which made the probe's orbit more prone to destabilization. The primary cause of destabilization was the existence of "mascons" ("mass concentrations") like those also present in our own Moon.
Mercury's proximity to the Sun (resulting in stronger solar tidal forces and solar radiation acting upon the probe) was a secondary factor - not entirely negligible, but not the chief factor.
If Messenger had been placed into a circular orbit around Mercury, and if Mercury were a smooth, uniform sphere with an even distribution of mass, and hence no variations in its gravity field, Messenger would have remained in orbit for decades and perhaps centuries.
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