Is there a strongly solved game with infinite possible board arrangements?
Yes, with Nim being the best-known example
You ask for a strongly solved game (presumably referring to the term in combinatorial game theory). According to Wikipedia, games that are strongly solved

Provide an algorithm that can produce perfect moves from any position, even if mistakes have already been made on one or both sides.

It goes on to note that in many cases, this is determined by brute force:

By contrast, &quot;strong&quot; proofs often proceed by brute force—using a computer to exhaustively search a game tree to figure out what would happen if perfect play were realized. The resulting proof gives an optimal strategy for every possible position on the board.

For instance, we can strongly solve 3x3 tic-tac-toe because we can brute force every possible combination and determine perfect moves. Of course, brute force is impossible for an infinite board size, and some games are only strongly solved up to a certain board size. For instance, Hex is only strongly solved up to a 6x6 board. Still, the hope is that this brute force on games like these will lead to an algorithm for perfect play that works for an infinite size (e.g. an NxN board).
The best studied strongly solved game is Nim, where an algorithm has been found that produces perfect play from any position of any size heaps and objects. This game can be played on a board (or just by drawing on a piece of paper), so it counts as a board game.
Most other combinatorial games are played on a fixed board and are not typically studied at NxN sizes. In some cases, an algorithm for those sizes may not be possible. As described above Hex has been strongly solved for a 6x6 board. However, an algorithm that works for NxN boards is unlikely to be found because the problem is PSPACE-complete, meaning that the number of possibles moves increases polynomially in relation to the size of the board, and it would take a polynomial amount of time to solve the game for a given size. It is suspected that this type of game is unsolvable because calculating for an infinite size would take a polynomially infinite amount of time.
Note that games like Chess, Checkers, and Go are EXPTIME-complete, which means that the number of possible moves increases exponentially in relation to the size of the board. Since an infinite board takes exponentially infinite time to solve, EXPTIME-complete problems like this one are also suspected to be unsolvable. Since exponential time takes longer than polynomial time, and PSPACE-complete problems are suspected to be unsolvable, EXPTIME-complete problems are generally considered to be completely unsolvable.
For reference, it took 18 years to strongly solve 8×8 checkers by brute force. It would take exponentially more work to solve a 9x9 board.
(It's been a long time since I've studied this stuff, but I vaguely recall some research considering what impact quantum computers might have on this type of problem. As I recall, it might be possible to reduce a PSPACE problem to a P problem, which can be efficiently solved in polynomial time regardless of the size of the input. That said, I'm not an expert in this field, so take it worth a grain of salt.)
Caveat: I'm not an expert in this field and it's been years since I've studied it. If I've made any incorrect statements about PSPACE-Complete etc. in this answer, please point it out in the comments or correct it.