Yes. In some cases, having a country specific top level domain (your "country code top-level domain", or ccTLD) of the country you are targeting can significantly increase your search engine rankings in that country.
But launching a new domain for a different market isn't simple. There are several things that can trip you up, including:
What Happens When You Launch a New ccTLD Domain?
What we would expect to happen with the launch of a new site is that your new site will start to rank, slowly creep up the results, and then overtake your existing sites' rankings. Eventually, your non-specific ccTLD will drop down in rankings in that country, but that's OK because the new one will outrank it.
One of our clients has franchises all over North America. Hundreds in the States and just a handful in Canada. Because Canada is such a small market in comparison to the US, they were servicing both marketing through a .com site. It performed well and was generating solid leads, but we wanted to take it one step further.
The .ca site needs to be different than the .com site. The audiences are similar, but their needs are different. The imagery, spelling, user benefits and even language used on the site should be reviewed and revised. For this client we integrated French versions of all the pages, which is not a feature of the .com site.
While site launch seems like it would be the last step, it really isn't. A soft-launch in your new market allowed the Canadian franchises and select focus groups a chance to use the site and identify any issues that the American audience had not previously encountered.
Your existing site has had a few years to gain traction in this market and you're new site will not take it's place overnight. This is especially true if you have location-specific pages. You've put in the time to create a market-specific user experience so make sure you're redirecting the location-specific pages on your old site to the new site.
For example: domain.com/showrooms-calgary should be 301'd to domain.ca/showrooms/calgary
Having the redirect in place is helpful to get your users to the right page, but to help your new site gain some authority and ranking power in the market you are targeting, you'll want to update your links to those pages. Your business listings on Yelp and other directories, your social media profiles, etc. should all be revised to point to the site that is specific to their audience.
It will take a few weeks to start to see the changes but when you notice that your new ccTLD domain is moving up the rankings and then outranking the existing site, it's very rewarding. Track both domains separately and make sure you're tracking the right search engine for each domain (tracking domain.ca on Google CA and domain.com on Google US). See which pages are performing well, and which ones are lagging, and then continue to optimize accordingly.
With some planning, a good execution, and lots of tweaks, you can skyrocket your localized rankings with a ccTLD.