Here is one piece of evidence which says no. The bottom line seems to be that large banks are better at mobilizing capital internationally:
An essential step to see if this view has ground is analysing how multinational banks have been faring in the months of the crisis, in comparison to domestic banks. Given the key role of these multinational banks in supporting the real economy, whether their subsidiaries in host countries have been restraining the amount of resources provided to local residents during the recession.
In a recent paper prepared for the Economic Policy panel in Madrid (Barba Navaretti et al. 2010), we provide both aggregate and micro evidence that multinational banks have actually contributed to the stabilisation of financial markets in Europe and to the provision of credit to the real economy (notwithstanding the systemic nature of the crisis).
The aggregate banking statistics of the Bank for International Settlements show that the total local claims held by the affiliates (branches and subsidiaries) of multinational banks in host countries in Europe have been stable and even increasing between the beginning of 2007 and the third term of 2009, both in absolute terms and normalised by GDP or total financial activities (Figure 1 for total local claims in local currency). Claims have been rising also in the major Central and Eastern European Countries where foreign banks account for the dominant share of total banking activities. This pattern is consistent with broader worldwide evidence.
Here is the conclusion:
Contrary to popular belief, our results show that a world with only small and domestic banks is no safer. Rather, we show that the size and the global extensions of the activities of multinational banks positively contributed to hedging the downturns of the crisis, even in transition economies.