Massive MIMO uses many antenna elements at the base station, and a simple antenna at the user equipment. Which means that the base station aims its signal at the user equipment by appropriately adjusting the relative phase of the signal transmitted by, or received from, each antenna element.

It therefore seems inescapable that fading which affects only some of the antenna elements will disrupt the base station's aim, and would likely result in loss of connectivity.

The assumption of large scale fading considered independent of the M antennas is quite realistic for current cellular Macro cell type of implementations. If the large scale fading is different per antenna, we can think of the antenna elements to belong to different terminals, i.e. a distributed MIMO communications. In that case, the effect of shadowing , as an example, can be lumped in the channel model  by multiplying by a diagonal matrix. The elements of this matrix represent the shadowing process per antenna, see http://www.jocm.us/uploadfile/2013/0927/20130927031847665.pdf for the mathematical treatment. As long as the small and large scale fading effect can be estimated at the Rx and TX, then suitable link adaption (power per antenna and spatial shaping matrix) can be implemented. No loss of connectivity would be expected in this case. 

The scenario with different large scale fading has been studied in several papers, where some consider general channel covariance matrices:

J. Hoydis, S. ten Brink, and M. Debbah, “Massive MIMO in the UL/DL of cellular networks: How many antennas do we need?,” vol. 31, no. 2, pp. 160–171, Feb. 2013.

And other papers focus on distributed or partially distributed antenna deployments:

Emil Björnson, Michail Matthaiou, Mérouane Debbah, “Massive MIMO with Non-Ideal Arbitrary Arrays: Hardware Scaling Laws and Circuit-Aware Design,” IEEE Transactions on Wireless Communications, vol. 14, no. 8, pp. 4353-4368, August 2015.

Hien Quoc Ngo, Alexei Ashikhmin, Hong Yang, Erik G. Larsson, and Thomas L. Marzetta, "Cell-Free Massive MIMO: Uniformly great service for everyone," IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Stockholm, Sweden, June 2015.

By distributing the antennas of the BS you will get less channel hardening (the gain of the effective channel will fluctuate more due to small-scale fading) but you might gain a lot in terms of macro-diversity (it is more probably that some of the antennas are close to the user).