Can I Ignore Him?

[Margo]

I'd reply back with something like "I appreciate the Link connect and will be in touch if our company is in need of (type of business) services.  Right now, we are all set.  Thank you." and hopefully he'll take the hint you aren't interested.

This. I think that the invite to coffee may have sent him a bit of a mixed message, although clearly he is a little clueless not to have tkaen the hint with your first response.

[Barney girl]

I still take the view that in business circles in the UK this is not sending out a mixed message. The connection on LinkedIn would have had to be either naive or deliberately obtuse to think of it as a social invitation. He was from a local firm of solicitors and they should be training their staff better on how to do business networking. If the OP or any of her colleagues have other connections with the firm it might be as we'll to drop a word in the ear of one of the partners. I know we'd be appalled if one of our staff did this as it could affect the reputation of the firm.