Is it possible for a natural fluorescent mineral to glow from the sun's rays?
The lifetime of a fluorescence excited state is on the nanosecond to microsecond time scale.  So once the excitation source light is removed, the emission of light will stop within microseconds. 

Note that fluorescence is distinct from phosphoresence, and phosphorescence can involve longer lifetime excited states.

Spectroscopic Characterization of the Uranium Carbonate Andersonite provides quantitative information on fluorescence lifetimes for Andersonite, and lifetimes are in the microsecond range.  

Regarding phosphorescence, see Phosphorescence; or, The emission of light by minerals, plants, and animals (1862):


  SEVERAL substances manifest the strange property  of emitting light
  when they are placed in darkness,  after having been exposed for some
  time to the  direct rays of the sun. In some cases a very short 
  exposure to sunlight is sufficient to excite the  manifestation of
  this remarkable property, and in  others the direct rays of the sun
  are not necessary :  it suffices that the substance experimented upon 
  be exposed to the dull light of a cloudy day. To  this phenomenon the
  denomination of Phosphores-  cence after insolation has been given. 
  
  ' The substances which possess this property in  the highest degree
  are the Bologna stone, or solar  phosphorus, certain varieties of
  fluor-spar and  carbonate of lime, some fossils, calcined shells or 
  pearls, phosphate of lime, arseniate of lime, etc.  Many diamonds
  shine with brilliancy in the dark  if they have previously undergone
  an exposure of  some seconds' duration only to solar light. But  no
  substance surpasses in this respect sulphuret  of barium. 
  
  It is now a long time since the cobbler of Bologna,,  in Italy,
  astonished and amused his friends with a  peculiar substance since
  known as Bologna phosphorus, Bologna stone, or Solar phosphorus,
  which  shines brightly in the dark after having been placed  in the
  sunlight for some time. This substance is  sulphuret of barium. The
  cobbler prepared it by  heating red-hot with charcoal a piece of
  sulphate  of baryta, or Barytine, (Fig. I,) a stone which he...