The pseudorapidity density of charged particles, Formula omitted, in p-Pb collisions has been measured at a centre-of-mass energy per nucleon-nucleon pair of Formula omitted = 8.16 TeV at mid-pseudorapidity for non-single-diffractive events. The results cover 3.6 units of pseudorapidity, Formula omitted. The Formula omitted value is Formula omitted at Formula omitted. This quantity divided by Formula omitted / 2 is Formula omitted, where Formula omittedis the average number of participating nucleons, is 9.5% higher than the corresponding value for p-Pb collisions at Formula omitted = 5.02 TeV. Measurements are compared with models based on different mechanisms for particle production. All models agree within uncertainties with data in the Pb-going side, while HIJING overestimates, showing a symmetric behaviour, and EPOS underestimates the p-going side of the Formula omitted distribution. Saturation-based models reproduce the distributions well for Formula omitted. The Formula omitted is also measured for different centrality estimators, based both on the charged-particle multiplicity and on the energy deposited in the Zero-Degree Calorimeters. A study of the implications of the large multiplicity fluctuations due to the small number of participants for systems like p-Pb in the centrality calculation for multiplicity-based estimators is discussed, demonstrating the advantages of determining the centrality with energy deposited near beam rapidity.