The BSA Airsporter was the premier line of airguns made by BSA of Birmingham England, back in the good ol' days. Every part is still available here in UK from a number of suppliers, mainly John Knibbs.
The 'Pylarm' pellets were also made by BSA specially for their range of medium quality airguns, mainly for small pest control and leisurely target shooting. The name is a take on the BSA company logo of three stacked Martini rifles - known in military circles as 'piled arms', hence the word.
Yours appears to be in fine fettle and should, with some care, give many years of sterling service. It was made between 1959 and 1965 -
https://www.airgunspares.com/whats-my-bsa
Note that in UK, the maximum legal m/e for any air rifle is 12 ft lbs - any higher than that and it's classed as a regular firearm and needs the FAC - equivalent in many respects to the Canadian PAL.
Please do NOT try to uprate it - it really is not intended to drive holes through one side of a deer and out the other.
It's hard for me to give a value as I live in UK, but they can vary by condition and age - older and better = more money, obviously. The newer German-made versions, with plastic sights, are not much sought-after - it is the originals from the late forties that attract the higher prices, often up to CAN$750 or even more.