Possibly, but can't speak for this specific starter.
Some/most Fiat starters have a bolted on solenoid and a
two winding pull in. Key energises first coil, solenoid moves
forward and activates main contacts, plus a second helper
coil to put extra pressure on the main heavy duty contacts.
It may be the solenoid is not moving freely keeping the
main contacts closed, or, possibly you have an ignition
switch/circuit  problem, with a leakage or short, equivalent
to keeping the key turned to the spring loaded start position.
Wiring a 12V bulb across the feed from the ignition switch
to chassis will tell you which is the case. Note the starter
has a feed direct from the battery to the heavy duty lug.
It makes a nasty splat if you unbolt it and it shorts to
chassis. You could also have a short or insulation breakdown
between the big from battery lug, and the smaller from
ignition switch, push on blade. If the starter stops when
you remove the small blade then it's probably a problem
at the ignition switch end, if it keeps running at the starter
end.
Cost of repair depends whether you go to a rip-off merchant
(main dealers also expensive) or a small local well
recommended, self employed auto electrician. Replacing
either the starter, or ignition switch, is over a hundred
quid with labour and VAT. Removing and servicing a
starter (which few would do nowdays) just under the
hundred. Replacement starter from a breakers yard
about 25-30 quid.