Infant mortality rates in 34 Sub-Saharan African countries (1960-2016), obtained from the Federal Reserve Bank of St. Louis database, were examined in this paper by focusing on the degree of persistence and non-linearities in the growth rate series. Persistence deals with the degree of association between the observations. Non-linearity occurs when departing from the linear assumption as in a time trend. These two issues are relevant in this context because they are intimately related. Based on the high degree of persistence observed in the series examined, instead of investigating structural breaks, which produce abrupt changes in the data, a non-linear approach was used based on Chebyshev polynomials in time, producing smooth rather than abrupt changes. This approach has never been examined in a unified framework in the treatment of infant mortality rates. The results indicate that half of the countries examined display non-linearities and the orders of integration of the series are extremely large in all cases, being around two in the majority of them. Looking at the growth rate series, significant negative trends were observed for: Chad, Equatorial Guinea and Mozambique. Evidence of mean reversion and thus transitory shocks, were observed for Lesotho, Rwanda, Botswana and Mozambique. Time dynamics of the series were expected to persist in order to ascertain the decline in mortality rates. Therefore, serious government interventions are required in managing infant health in these countries.