Yes, FLHTCUI has it right. It's an early FN-manufactured semi-automatic .22LR (also available in .22 Short) using the Browning design. Truly a classic rifle given its slick design, light weight and lively handling, and longevity of production--now over 100 years, with half a million sold since 1914. FN began manufacturing these in 1914, so yours could be that old--possibly made in the 1920s. This early version had a magazine loading port on the top of the stock, whereas the later ones had the port on the right side of the buttstock. I lucked into a like-new used one decades ago. This was when they had the pictured splinter forend (I think mine was even narrower), before they branded the rifle as a Browning (rather than FN) and put on a much fuller hand-filling forend, altered the half pistol grip to a full PG, and checkered the pistol grip and forend. Same original John Browning semi-auto action, though.
Along the way Remington got US manufacturing rights (presumably licensed from Browning) from 1922 to 1949, labeling their version the Model 24 and later 241. I'm assuming that FN continued production in Belgium in those years for sale in Europe and possibly other countries (maybe Canada), although perhaps someone can confirm this.
Here's a link to a brief description of the evolution of this rifle, along with another link inside to the FN Factory Manual for the rifle. You'll have to copy and paste it into your browser:
https://sites.google.com/site/hbsauk/knowledge-and-research/articles/fn-browning-22-calibre-automatic-rifle/
It's hard to put a value on that specimen, what with all the rust, missing front sight, and poor-quality and banged-up wood. The later Browning models in 90 to 95% condition--particularly those that were still made in Belgium--seem to go for around $500-$650 these days in Canada, if I've been reading the ads right, so you could use that as an upper-limit baseline for yours, which will obviously be worth much less if purchased for operational purposes. However, collectors might put a premium on such an old specimen, although I'm clueless as to what a reasonable "collector" price would be.