 Solution

False. The complete bipartite graph $K_{2009,2010}$ is $2009$-connected. Deleting fewer than $2009$ vertices leaves at least one vertex in each part and therefore a connected complete bipartite graph; deleting the smaller part disconnects the larger independent set. But its two parts have unequal sizes, so a matching cannot cover them both. Equivalently it has an odd number of vertices. Thus it has no perfect matching.